[Page] SIX LESSONS To the PROFESSORS of the MATHEMATIQUES, ONE OF GEOMETRY, THE OTHER OF ASTRONOMY: In the Chaires set up by the Noble and Lear­ned Sir HENRY SAVILE, in the University of Oxford.

LONDON. Printed by J. M. for Andrew Crook, at the Green-Dragon, in Pauls Church-yard.

To the Right Honourable, Henry Lord Pierrepont, Viscount Newarke, Earle of Kingstone, and Marquis of Dorchester.

My most Noble Lord,

NOt knowing on my own part any cause of the favour your Lordship has been pleased to express towards me, unless it be the Principles, Method, and Man­ners you have observed and approved in my Wri­tings; and seeing these have all been very much reprehended by men to whom the name of Publique Professors hath procured reputation in the University of Oxford; I thought it would be a forfeiture of your Lordships good opinion, not to ju­stifie my self in publique also against them. Which, whether I have sufficiently performed or not in the six following Lessons addressed to the same Professors, I humbly pray your Lordship to consider. The volume it self is too small to be offered to you as a Present; but to be brought before you as a Controversie it is perhaps the better for being short. Of Arts, some are demon­strable, others indemonstrable; and demonstrable are those the construction of the Subject whereof is in the power of the Artist himself; who in his demonstration does no more but deduce the Consequences of his own operation. The reason whereof is this, that the Science of every Subject is derived from a praecognition of the Causes, Generation, and Construction of the same; and consequently where the Causes are known, there is place for De­monstration; but not where the Causes are to seek for. Geome­try therefore is demonstrable; for the Lines and Figures from which we reason are drawn and described by our selves; and Ci­vill [Page] Philosophy is [...], we make the Common­wealth our selves. But because of Naturall Bodies we know not the Construction, but seek it from the Effects, there lyes no demonstration of what the Causes be we seek for, but onely of what they may be.

And where there is place for Demonstration, if the first Prin­ciples, that is to say, the Definitions contain not the Generation of the Subject; there can be nothing demonstrated as it ought to be. And this in the three first Definitions of Euclide suffici­ently appeareth. For seeing he maketh not, nor could make any use of them in his Demonstrations, they ought not to be numbered among the Principles of Geometry. And Sextus Em­piricis maketh use of them (misunderstood, yet so understood as the said Professors understand them) to the overthrow of that so much renouned Evidence of Geometry. In that part therefore of my Book where I treat of Geometry, I thought it necessary in my Definitions to express those Motions by which Lines, Su­perficies, Solids and Figures were drawn and described; little expecting that any Professor of Geometry should finde fault therewith; but on the contrary supposing I might thereby not only avoid the Cavils of the Scepticks, but also demonstrate di­vers Propositions which on other Principles are indemonstrable. And truly, if you shall finde those my Principles of Motion made good, you shall find also that I have added something to that which was formerly extant in Geometry.

For first from the seventh Chapter of my Book de Corpere to the thirteenth, I have rectified and explained the Principles of the Science, id est, I have done that business for which Doctor Wallis receives the wages. In the seventh, I have exhibited and demonstrated the proportion of the Parabola and Parabo­lasters to the Parallelograms of the same height and base; which (though some of the propositions were extant without their demonstration) were never before demonstrated, nor are by any other then this method demonstrable.

In the eighteenth, (as it is now in English) I have demonstra­ted the (for any thing I yet perceive) Equation between the crook­ed line of a Parabola or any Parabolaster and a straight line.

[Page] In the twenty-third, I have exhibited the Center of Gravity of any Sector of a Sphere.

Lastly, the twenty-fourth, (which is of the nature of Refracti­and Reflexion) is almost all new.

But your Lordship will ask me what I have done in the twen­tieth, about the Quadrature of the Circle. Truely, my Lord, not much more then before. I have let stand there that which I did before condemn, not that I think it exact, but partly because the Division of Angles may be more exactly performed by it then by any organicall way whatsoever; and I have attempted the same by another Method, which seemeth to me very naturall, but of calculation difficult and slippery. I call them only Aggres­sions, retaining nevertheless the formall manner of Assertion used in Demonstration. For I dare not use such a doubtfull word as Videtur, because the Professors are presently ready to oppose me with a Videtur quod non. Nor am I willing to leave those Ag­gressions out, but rather to try if it may be made pass for law­full (in spight of them that seek honour not from their own per­formances but from other mens failings) amongst many difficult undertakings carryed through at once) to leave one and the great­est for a time behind; and partly because the method is such as may hereafter give further light to the finding out of the exact truth.

But the Principles of the Professors that reprehend these of mine, are some of them so void of sense, that a man at the first hearing, whether Geometrician or not Geometrician must abhor them. As for example;

  • 1. That two equall Proportions are not double to one of the same Proportions.
  • 2. That a Proportion is double, triple, &c. of a Number, but not of a Proportion.
  • 3. That the same Body, without adding to it, or taking from it, is sometimes Greater, and sometimes less.
  • 4. That a Quantity may grow less and less Eternally, so as at last to be equall to another Quantity; or (which is all one) that there is a Last in Eternity.
  • 5. That the nature of an Angle consisteth in that which lyes [Page] between the lines that comprehend the Angle in the very point of their concourse; that is to say, An Angle is the Superficies which lyes between the two Points which touch, or (as they un­derstand a Point) the Superficies that lyes between the two No­things which touch.
  • 6. That the Quo [...]ient is the Proportion of the Division to the Dividend.

Upon these and some such other Principles is grounded all that Doctor Wallis has said not onely in his Elenchus of my Geometry, but also in his Treatises of the Angle of Con­tact, and in his Arithmetica Infinitorum; which two last I have Fere in two or three leaves wholly and cleerly confuted. And I verily believe that since the beginning of the world there has not been, nor ever shall be so much absurdity written in Geometry, as is to be found in those books of his; with which there is so much presumption joyned, that an [...] of the like conjunction cannot be expected in less then a Platonick year. The cause whereof I imagine to be this, that he mistook the study of Sym­boles for the study of Geometry, and thought Symbolicall wri­ting to be a new kinde of Method, and other mens Demonstra­tions set down in Symboles new Demonstrations. The way of Analysis by Squares, Cubes, &c. is very anti­ent, and usefull for the finding out whatsoever is contained in the nature and generation of rectangled Plains (which also may be found without it) and was at the highest in Vieta; but I never saw any thing added thereby to the Science of Geometry, as be­ing a way wherein men go round from the Equality of rectan­gled Plains to the Equality of Proportion, and thence again to the Equality of rectangled Plains; wherein the Symboles serve only to make men go faster about, as greater Winde to a Winde-mill.

It is in Sciences as in Plants; Growth and Branching is but the Generation of the Root continued; nor is the Invention of Theoremes any thing else but the knowledge of the Constructi­on of the Subject prosecuted. The unsoundness of the Bran­ches are no prejudice to the Roots; nor the Faults of Theoremes to the Principles. And active Principles will correct false Theo­remes [Page] if the Reasoning be good; but no Logique in the world is good enough to draw evidence out of false or unactive Princi­ples. But I detain your Lordship too long. For all this will be much more manifest in the following Discourses: wherein I have not onely explained and rectified many of the most important Principles of Geometry: but also by the examples of those er­rors which have been committed by my Reprehenders, made manifest the evil Consequence of the Principles they now pro­ceed on. So that it is not only my own Defence that I here bring before you, but also a positive doctrine concerning the true Grounds, or rather Atomes of Geometry: which I dare only say are very singular: but whether they be very good or not, I submit to your Lordships judgement. And seeing you have been pleased to bestow so much time (with great success) in the read­ing of what has been written by other men in all kindes of Learn­ing. I humbly pray your Lordship to bestow also a little time upon the reading of these few and short Lessons; and if your Lordship finde them agreeable to your Reason and Judgement, let me (notwithstanding the clamour of my Adversaries) be con­tinued in your good opinion, and still retain the honour of being

My most Noble Lord,
Your Lordships most Humble and Obliged Servant. THOMAS HOBBES.


PAge 2. l. 11. for Art, [...]. act, l. 12. for [...], [...]. [...], p. 6. [...]. 23. for Mr. [...]. Sir, p. 21. l. 22. for Pr [...]position, r. Proportion, p. 55. l. 20. for Senctoribus, r. Senat [...]ribus, p. 56. l. 8. for Serberius, r. Sorberius.

[Page] [Page 1] LESSONS OF THE PRINCIPLES OF Geometry, &c.

To the egregious Professors of the Mathema­ticks, one of Geometry, the other of Astronomy, in the Cha [...]rs set up by the Noble and Learned Sir Henry Savile, in the University of Oxford.

I Suppose (most egregious Professors) you know already that by Geomerry (though the word import no more but the measuring of Land) is under­stood no less the measuring of all other Quantity, then that of Bodies. And though the Definition of Geometry serve not for proof, nor enter into any Geometricall Demonst [...]ation, yet for understanding of the Principles of the Science, and for a Rule to judge by, who is a Geometrician, and who is not, I hold it necessary to begin therewith.

Geometry is the Science of determining the quantity of any thing, not measured, by comparing it with some other Quantity, or Quantities measured. Which Science therefore whosoever shall go about to teach, must first be able to tell his Disciple what Measuring or Dimension is; by what each several kind of Quantity is Measured; what Quantity is, & what are the several kinds thereof. Therefore as they who handle any one part of Geometry, determine by Defiaition the signifie [Page 2] cation of every word whīch they make the Subject, or Praedicate of any Theoreme they under­take to demonstrate; so must he which intendeth to write a whole body of Geometry, Define and Determine the meaning of whatsoever word belongeth to the whole Science; The design of Euclid was to demonstrate the Properties of the five regular bodies mentioned by Plato; in which Demonstrations there was no need to alledge for Argument the Definition of Quantity, which it may be was the cause he hath not any where Defined it, but done what he undertook without it. And though having perpetually occasion to speak of measure, he hath not Defined Measure; yet instead thereof he hath in the beginning of his first Element, assumed an Axiome which serveth his turn sufficiently, as to the measure of lines, which is the eighth Axiome; That those things which lye upon one another all the way (called by him [...]) are equall. Which Axiome is nothing else but a description of the Art of Measuring Length, and Superficies. For this [...] can have no place in solid bodies, unless two bodies could at the same time be in one place. But amongst the Principles of Geometry universall, the Definiti­ons are necessary, both of Quantity and Dimension.

Quantity is that which is signified by what we answer to him that asketh, How much any thing is, and thereby determine the magnitude thereof. For magnitude being a word indefinite, if a man ask of a thing Quantum est, that is, How much it is; we do not satisfie him by say­ing it is magnitude or quantity, but by saying it is Tantum, so much. And they that first cal­led it in Greek [...], and in Latine Qua [...]tity, might more properly have called it, in La­tine Tantity, and in Greek [...]; and we, if we allowed our selves the Eloquence of the Greeks and Latines, should call it the So-muchness.

There is therefore to every thing concerning which a man may ask without absurdity how much it is, a certain Quantity belonging; determining the magnitude to be so much. Also wheresoever there is more and less, there is one kinde of Quantity or other. And first, there is the Quantity of Bodies, and that of three kindes; Length, which is by one way of Measuring; Superficies made of the complication of two Lengths, or the Measure taken, two wayes; and Solid which is the complication of three lengths, or of the measure taken three wayes; for breadth or thickness are but other Lengths. And the Science of Geometry so far forth as it con­templateth Bodies onely, is no more but by Measuring the length of one or more lines, and by the position of others known in one and the same Figure, to Determine by ratiocination, how much is the Superficies; and by Measuring Length, Breadth, and Thickness, to determine the Quantity of the whole Body. Of this kinde of Magnitudes and Quantities the Subject is Body.

And because for the computing of the Magnitudes of Bodies, it is not necessary that the Bo­dies themselves should be present (the Ideas and memory of them supplying their presence) we reckon upon those Imaginary Bodies, which are the Quantities themselves, and say the length is so great, the breadth so great, &c. which in truth is no better then to say the length is so long, or the breadth so broad, &c. But in the mind of an inteligent man it breedeth no mistake.

Besides the Quantity of Bodies, there is a Quantity of Time. For seeing men, without ab­surdity do ask how much it is; by answering Tantum, so much, they make manifest there is a a, quantity that belongeth unto Time, namely, a Length. And because Length cannot be an accident of Time, which is it self an accident, it is the accident of a Body; and consequently the length of the Time, is the Length of the Body; by which Length or Line, we determine how much the Time is, supposing some Body to be moved over it.

Also we not improperly ask with how much Swiftness a Body is moved; and consequently there is a Quantity of Motion or Swiftness, and the measure of that Quantity is also a line. But then again, we must suppose another motion, which determineth the time of the former. Also of Force there is a Question of How much, which is to be answered by So much; that is, by Quantity. If the Force consist in Swiftness, the Determination is the same with that of Swiftness, namely, by a Line; if in Swiftness and Quantity of Body joyntly, then by a Line and a Solid; or if in quantity of Body onely (as Weight) by a Solid onely.

[Page 3] So also is Number Quantity; but in no other sense then as a line is Quantity divided into equall parts.

Of an Angle, which is of two Lines whatsoever they be, meeting in one point the digression or openess in other points, it may be asked how great is that digression; This Que­stion is answered also by Quantity. An Angle therefore hath Quantity, though it be not the subject of Quantity; for the body onely can be the subiect, in which Body those [...]ing line are marked.

And because two lines may be made to divaricate by two causes; one, when having one end common, and immoveable, they depart one from another at the other ends circularly, and this is called surply an Angle; and the Quantity thereof is the Quantity of the Arch, which the two lines intercept.

The other cause is the bending of a straight line into a circular or other crooked line, till it touch the place of the same line, whilst it was straight, in one onely point. And this is called an Angle of contingence. And because the more it is bent, the more it digresseth from the Tangent, it may be asked how much more? and therefore the answer, must be made by Quan­tity; and consequently an Angle of Contingence hath its Quantity as well as that which is cal­led simply an Angle. And in case the digression of two such crooked lines from the Tangent be uniform as in Circles, the Quantity of their digression may be determined. For if a straight line be drawn from the point of Contact, the digression of the lesser Circle will be to the digres­sion of the greater Circle, as the part of the line drawn from the point of Contact, and inter­cepted by the Circumference of the greater Circle is to the part of the same line intercepted by the Circumference of the Lesser Circle, or, which is all one, as the greater Radius is to the les­ser Radius. You may guess by this what will become of that part of your last Book, where you handle the Question of the Quantity of an Angle of Contingence.

Also there lyeth a Question of how much Comparatively one magnitude is to another mag­nitude, as how much water is in a Tun in respect of the Ocean, how much in respect of a Pi [...]; little in the first respect, much in the Latter. Therefore the Answer must be made by some re­spective Quantity. This respective Quantity is called Ratio and Proportion, and is determi­ned by the Quantity of their differences; and if their differences be compared also with the Quantities themselves that differ, it is called simply Proportion, or Proportion Geometricall. But if the differences be not so compared, then it is called Proportion Arithmeticall. And where the difference is none, there is no Quantity of the Proportion, which in this case is but a bare comparison.

Also concerning Heat, Light, and divers other Qualities, which have degrees, there lyeth a question of how much, to be answered by a so much, and consequently they have their Quan­tities, though the same with the Quantity of Swiftness: because the intensions and remissi­ons of such Qualities are but the intensions and remissions of the Swiftness of that motion by which the Agent produceth such a quality. And as Quantity may be considered in all the ope­rations of Nature, so also doth Geometry run quite thorow the whole body of Naturall Philosophy.

To the Principles of Geometry the definition appertaineth also of a M [...]asure, which is this, One Quantity is the Measure of another Quantity, when it, or the Multiple of it, is Coi [...] ­cident in all points, with the other Quantity. In which Definition in stead of that [...] of Euclid, I put Coincidence. For that superposition of Quantities by which they ren­der the word [...] cannot be understood of Bodies, but only of Lines and Superficies. Ne­vertheless many Bodies may be Coincident successively with one and the same place, and that place will be their Measure, as we see practised continually in the measuring of Liquid Bodies, which Art of M [...]asu [...]ing may properly be called [...], but not Superposition.

Also the definitions of Greater, Less, and Equall, are necessary Principles of Geometry. For it cannot be imagined that any Geometrician should demonstrate to us the Equality, and In [...] ­quality [Page 4] of magnitudes, except he tell us first what those words do signifie. And it is a wonder to me' that Euclide hath not any where defined what are Equals, or at least, what are Equall Bodies, but serveth his turn throughout with that forementioned [...], which hath no place in Solids, nor in Time, nor in Swiftness, nor in Circular, or other crooked lines; and therefore no wonder to me, why Geometry hath not proceeded to the calculation neither of crooked lines, nor sufficiently of Motion, nor of many other things, that have proportion to one another.

Equall Bodies, Superficies, and Lines are those of which every one is capable of being co-inci­dent with the place of every one of the rest. And Equall Times wherein with one and the same Motion Equall lines are described. And Equally Swift are those Motions by which we run over equall spaces in any time determined by any other motion. And universally all Quanti­ties are Equall, that are measured by the same number of the same Measures.

It is necessary also to the Science of Geometry, to define what Quantities are of one and the same kind, which they call Homogeneous; the want of which definitions hath produced those wranglings (which your Book De Angulo Contactus will not make to cease) about the Angle of Contingence.

Homogeneous quantities are those which may be compared by ( [...]) application of their Measures to one another; So that Solids and Superficies, are Heterogeneous quantities, because there is no coincidence or application of those two dimensions.

No more is there of Line and Superficies, nor of Line and Solid, which are therefore Hetero­geneous. But Lines and Lines, Superficies and Superficies, Solids and Solids, are Homogeneous.

Homogeneous also are Line, and the Quantity of Time; because the Quantity of Time is measured by application of a Line to a Line; for though Time be no Line, yet the Quantity of Time is a Line, and the length of two Times is compared by the length of two Lines.

Weight, and Solid have their Quantity Homogeneous, because they measure one another by application, to the beam of a Balla [...]. Line and Angle simply so called have their Quantity Homogeneous, because their measure is an Arch or Arches of a Circle applicable in every point to one another.

The Quantity of an Angle sinply so called, and the Quantity of an Angle of Contingence are Heterogeneous. For the measures by which two Angles simply so called are compared, are in two coincident Atches of the same Circle; but, the measure by which an Angle of Coatin­gence is measured, is a straight line intercepted between the point of Contact, and the Circumfe­rence of the Circle; and therefore one of them is not applicable to the other; and consequently, of these two sorts of Angles, the Quantities are Heterogeneous. The Quantities of two Angles of Contingence are Homogeneous; for they may be measured by the [...] of two Lines, whereof one extream is common, namely, the point of Contact, the other Extreams, are in the Arches of the two Circles.

Besides this knowledge of what is Quantity and Measure, and their severall sorts, it behoveth a Geometrician to know why, and of what they are called Principles. For not every Proposi­tion that is evident, is therefore a Principle. A Principle is the beginning of something. And because Definitions are the beginnings or first Propositions of Demonstration, they are there­fore called Principles, Principles, I say, of Demonstration. But there be also necessary to the teaching of Geometry other Principles, which are not the beginnings of Demonstration, but of Construction, commonly called Petitions; as that it may be granted that a man can draw a straight Line, and produce it; and with any Radius; on any Center describe a Circle, and the like. For that a man may be able to describe a square, he must first be able to draw a straight line; and before he can describe an Aequilaterall Triangle, he must be able first to de­scribe a Circle. And these Petitions are therefore properly called Principles (not of Demon­stration, but) of Operation. As for the commonly received third sort of Principles, called Common Notions, they are Principles, onely by permission of him that is the Disciple [...]; who being [...]nuous, and comming not to cavill but to learn, is content to receive them (though [Page 5] Demonstrable) without their Demonstrations. And though Definitions be the onely Principles of Demonstration, yet it is not true that every Definition is a Principle. For a man may so precisely determine the signification of a word, as not to be mistaken, yet may his Definition be such, as shall never serve for proof of any Theoreme, nor ever enter into any demonstration (such as are some of the Definitions of Euclide) and consequently can be no beginnings of De­monstration, that is to say, no Principles.

All that hitherto hath been said, is so plain and easie to be understood, that you cannot (most Egregious Professors) without discovering your ignorance to all men of reason, though no Geometricians, deny it. And the same (saving that the words are all to be found in Dicti­onaties) new; also to him that means to learn, not onely the Practice, but also the Science of Geometry necessary, and (though it grieve you) mine. And now I come to the Definitions of Euclide.

The first is of a Point. [...] &c. Signum est, eujus est pars nulla, that is to say, a. Marke is that of which there is no part. Which definition, not onely to a candid, but also to a rigid construer is sound and usefull. But to one that neither will interpret candidly, nor can interpret accurately, is neither usefull nor true. Theologers say the Soul hath no part, and that an Angel hath no part, yet do not think that Soul or Angel is a point. A mark, or as some put instead of it [...], which is a mark with a hot Iron, is visible; if visible then it hath Quantity, and consequently may be divided into parts innumerable. That which is indivisible is no Quantity; and if a point be not Quantity, seeing it is neither substance nor Quality, it is nothing. And if Euclide had meant it so in his definition, (as you pretend he did) he might have defined it more briefly, (but ridiculously) thus, a Point is nothing. Sir Henry Savile was bet­ter pleased with the Candid interpretation of Proclus, that would have it understood respectively to the matter of Geometry. But what meaneth this respectively to the matter of Geometry? It meaneth this, that no Argument in any Geometricall demonstration should be taken from the Division, Quantity, or any part of a Point; which is as much as to say, a Point is that whose Quantity is not drawn into the demonstration of any Geometricall conclusion; or (which is all one) whose Quantity is not considered.

An accurate interpreter might make good the definition thus, a Point is that which is undi­vided; and this is properly the same with cujus non est pars: for there is a great difference be­tween undivided, and in [...]ivisible, that is, between cujus non est pars, and cujus non potest esse pars. Division is an Act of the understanding; the understanding therefore is that which maketh parts, and there is no part where there is no consideration bat of one. And consequently Euclides definition of a Point, is accurately true, and the same with mine, which is, that a Point is that Body whose Quantity is not considered. And considered, is that (as I have defined it, Chap. 1. at the end of the third Article) which is not put to account in demonstration.

Euclide therefore seemeth not to be of your opinion, that say a Point is nothing. But why then doth he never use this definition in the Demonst [...]ation of any Proposition? Whether he useth it expressly or no, I remember not; but the 16th. Proposition of the third Book without the force of this definition is undemonstrated.

The second Definition is of a Line. [...]. Linea est longitudo latitudinis expers, a Line is length which hath no breadth; and if candidly interpreted, sound enough, though rigorously, not so. For to what purpose is it to say Length not Broad, when there is no such thing as a broad length. One Path may be broader then another Path, but not one Mile then another Mile; and it is not the Path but the Mile which is the ways Length. If therefore a man have any ingenuity he will understand it thus, That' a Line is a Body whose length is considered without its breadth, else we must say absurdly a broad length; or untruly, that there be bodies which have length and yet no breadth; and this is the very sense which A­pollonius (saith Proclus) makes of this Definition; when we measure, sayes he, the length of a way, we take not in the breadth or depth, but consider onely one Dimension. See this of [Page 6] Proclus cited by Sir Henry Saviles where you shall finde the very word consider.

The fourth Definition is of a straight line, thus [...], &c. Recta linaa est quae ex aequo sua ipstus puncta inter jacet. A straight line is that which lieth equally (or per­haps evenly) between its own Points. This Definition is inexcusable. Between what Points of its own can a straight line lye but between its extreams? And how lies, it evenly between them, unless it swarve no more from some other line which hath the same Extreams, one way then ano­ther? And then why are not between the same Points both the lines straight? How bitterly, and with what insipide justs, would you have reviled Euclide for this, if living now he had written a Loviathan. And yet there is somewhat in this Definition to help a man, not onely to conceive the nature of a straight line (for who doth not conceive it?) but also to express it. For he meant perhaps to call a straight line that which is all the way from one Extream to another, equally di­stant from any two or more such Lines as being like and equall have the same Extreams. So the Axis of the Earth is all the way equally distant from the circumference of any two or more Meri­dians. But then before he had defined a straight line, he should have defined what lines are like, and what are equall. But it had been best of all, first to have defined crooked lines, by the possi­bility of a diduction or setting further asunder of their extreams; and then straight Lines, by the impossibility of the same.

The seventh Definition, which is that of a plain Superficies, hath the same fault.

The Eighth is of a plain Angle. [...], &c. Angulus Planus est duarum Linearum in plano se mutuo tangentium, & non indirectum jacentium, al­terius ad alteram inclinatio. A Plain Angle is the inclination one towards another of two lines that touch one another in the same Plain, and lye not in the same straight line: Besides the faults here observed by Mr. Henry Savile, as the clause of not lying in the same straight line, and the obscurity or aequivocation of the word inclination, there is yet another, which is, that by this Definition two Right Angles together taken, are no Angle; which is a fault which you some­where (asking leave to use the word Angle [...], acknowledge, but avoid not. For in Geometry, where you confesse there is required all possible accurateness, every [...] is a fault. Besides you see by this Definition, that Euclide is not of your, but of Clavius his opinion. For it is manifest that the two lines which contain an Angle of Contact, incline one towards another, and come together in a Point, and lye not in the same streight line, and conse­quently make an Angle.

The thirteenth Definition is exact, but makes against your Doctrine, that a Point is nothing. Examine it. [...]. Terminus est quod alicujus extremum est. A Term or Bound is that which is the Extream of any thing We had before, The extreams of a Line are Points. But what is in a Line the extream, but the first or last part, though you may make that part as small as you will? A point is therefore a part, and nothing is no extream.

The fourteenth, [...] Figura est (subaudi Quantitas) quae ab aliquo, vel aliquibus terminis undi (que) continetur five clauditur. A Fi­gure is Quantity contained within some bound or bounds. Or shortly thus, A Figure is Quantity every way determined, is in my opinion as exact a Definition of a Figure as can possibly be given, though it must not be so in yours. For this determination is the same thing with circum­scription; and whatsoever is any where (U B Icun (que)) definitivè is there also circumscriptivè; and by this means, the distinction is lost, by which Theologers, when they deny God to be in any place, save themselves from being accused of saying he is no where; for that which is no where is nothing. This definition of Euclide cannot therefore possibly be embraced by you that carry double, namely, Mathematicks and Theology. For if you reject it, you will be cast out of all Mathematick Schooles; and if you maintain it, from the Society of all School-Divines, and lose the thanks of the Favour you have shewn (you the Astronomer) to Bishop Bramhall.

[Page 7] The fifteenth is of a Circle, [...], &c. A Circle is a plain Fi­gure comprehended by one line which is called the Circumference, to which Circumference all the straight lines drawn from one of the points within the Figure, are equall to one another. This is true. But if a man had never seen the generation of a Circle by the motion of a Compass or other aequivalent means, it would have been hard to perswade him, that there was any such Fi­gure possible. It had been therefore not amiss first to have let him see that such a Figure might be described. Therefore so much of Geometry is no part of Philosophy, which seeketh the proper passions of all things in the generation of the things themselves.

After the fifteenth till the last or thirty fifth Definition all are most accurate, but the last, which is this, Parallel straight lines are those which being in the same Plane, though infinitely pro­duced both wayes shall never meet. Which is lesse accurate. For how shall a man know that there be straight lines, which shall never meet, though both wayes infinitely produced? Or how is the Definition of Parallels, that is, of lines perpetually aequidistant good, wherein the nature of aequi­distance is not signified? Or if it were signified, why should it not comprehend, as well the Paral­lelism of Circular and other crooked lines, as of streight, and as well of Superficies, as of lines? By Parallels is meant aequidistant both Lines and Superficies, and the word is therefore not well defi­ned without defining first equality of distance. And because the distance between two lines or Superficies, is the shortest line that can joyn them, there either ought to be in the definition, the shortest distance, which is that of the Perpendicular, and without inclination, or the distance in equall inclination, that is, in equall Angles. Therefore if Parallels be defined to be those Lines or Superficies, where the Lines drawn from one to an other in equal Angles be equal, the Definition, as to like Lines, or like Superficies, will be Universall and Convertible. And if we add to this Defini­tion, that the equall Angles be drawn not opposite wayes, it will be absolute, and Convertible in all Lines and Superficies; and the definition will be this, Parallels are those Lines and Superficies between which every line drawn, in any Angle, is equall to any other line drawn in the same Angle the same way. For by this Definition the distance between them will perpetually be equall, and consequently they will never come nearer together, how much, or which way soever they be produced. And the converse of it will be also true, If two Lines, or two Superficies be Paralle, and a straight line be drawn from one to the other, any other straight line, drawn from one to the other in the same Angle, and the same way, will be equall to it. This is ma­nifestly true, and (most egregious Professors) new, at least to you.

And thus much for the Definitions placed before the first of Euclides Elements.

Before the third of his Elements is this Definition. In circulo aequaliter distare à centro rectae lineae dicuntur, cum prependiculares quae à centro in ipsas ducuntur sunt [...] aequales. In a circle two straight Lines are said to be equally distant from the Center, upon which the perpen­diculars drawn from the Center are equall. This is true; but it is rather an Axiome then a Definition, as being demonstrable that the Perpendicular is the measure of the distance between a Point and a straight or crooked Line.

Before the fifth Element the first Definition is of a Part. Pars est magnitudo magnitudinis, minor majoris, cum minor metitur majorem. A Part is one magnitude of another, the [...]esse of the greater, when the lesse measureth the greater. From which Definition it followeth, that more then the half is not a part of the whole. But because Euclide meaneth here an Aliquot part, as a half, a third, or a fourth, &c. It may pass for the Definition of a measure under the name of part; as thus, a Measure is a Part of the whole, when multiplyed, it may be equall to the whole, though properly a Measure is externall to the thing measured, and not the Aliquot part it self; but equall to an Aliquot part.

But the third Definition is intollerable, It is the definition of [...], in Latin Ratio, in English Proportion, in this manner; [...]. Ratio est duarum magnitudinum ejusdem generis mutua quaedam secun­dum quantitatem babitudo. Proportion is a certain mutuall habitude in Quantity, of two mag­nitudes [Page] of the same kinde, one to another. First, we have here ignotum per ignotīus; for every man understandeth better what is meant by Proportion then by Habitude. But it was the phrase of the Greeks when they named like Proportions, to say, the first to the second [...], id est, Ita se habet, and in English, is as, the third to the fourth. As for Example (in the Propor­tions of two to four, and three to six) to say two to four [...], id est, ita se habet, id est, is as, three to six. From which phrase Euclide made this his Definition of Proportion by [...], which the Latines translate quaedam habitudo. Quaedam in a Definition is a most certain note of not understanding the word defined; And in Greek [...] is much worse; for to render rightly the Greek definition, we are to say in English, that Proportion is a what­shall-I-call-it isaesse, or soness of two magnitudes &c. Then which nothing can be more un­worthy of Euc [...]ide. It is as bad as any thing was ever said in Geometry by Orontius, or by Dr. Wallis, That Proportion is Quantity compared, that is to say, little or great in respect of some other Quantity (as I have above defined it) is I think intelligible.

The fourth is, [...]. Proportio verò est rationum similitudo. Here we have no one word by which to render [...]; for our word Proportion, is already bestowed upon the rendring of [...] Neverthelesse the Greek may be translated into English thus, Iterated Proportion is the similitude of Proportions. But Iterated Proportion is the same with eadem Ratio. To what purpose then serveth the sixth Definition, which is of eadem Ratio? For [...] and eadem Ratio, and Similitudo Rationum, are the same thing, as appeareth by Euclide himself; where he defines those Quantities, that are in the same Proportion by [...]. Therefore the sixth Definition is but a Lemma, and assumed without demonstration.

The fourteenth, Compositio Rationis est sunptio Antecedentis cum Consequente, ceu unius, ad ipsum consequentem. To Compound Proportion, is to take both Antecedent, and Conse­quent together, as one magnitude, and compare it to the consequent, Is good; though he might have compared it as well with the Antecedent; For both wayes it had been a Composition of Proportion. We are to note here, that the Composition defined in this place by Euclide, is not adding together of Proportions, but of two Quantities that have Proportion. And therefore it is not the same Composition which he defineth in the fourth place before the sixth Element; for there

He defineth the addition of [...] Proportion to another Proportion in this manner [...], &c. A Proportion is said to be compounded of Proportions, when their Quantities multiplye [...] into one another make a Proportion; as when we would compound or add together, the Proportions, of three to two, and of four to five, we must multiply three and four, which maketh twelve, and two and five which maketh ten. And then the Proportion of twelve to ten, is the sum of the Proportions, of three to two, and of four to five; which is true; but not a de­finition; for it may and ought to be demonstrated. For to define what is addition of two Pro­portions (which are alwayes in four Quantities, though sometimes one of them be twice named) we are to say, that they are then added together when we make the second to another in the same Proportion, which the third hath to the fourth.

And thus much of the Definitions; of which, some, very few, you see are faulty; the rest ei­ther accurate, or good enough if well interpreted. For before the rest of the Elements all are accu­rate, notwithstanding, that you allow not for good any definition in Geometry that hath in it the word motion: of which there be divers before the Eleventh Element. But I must here put you in minde, that Geometry being a Science, and all Science proceeding from a precognition of causes, the definition of a Sphere, and also of a Circle, by the generation of it, that is to say, by motion, is better then by the equality of distance from a Point within.

[Page 9] The second sort of Principles, are those of Construction, usually called Postulata, or Peti­tions. For as for those notiones communes, called Axiomes, they are from the definitions of their terms demonstrable, though they be s [...] evident as they need not demonstration. These Petitions are by Euclide called [...], such as are granted by favour, that is, simply Peti­tions whereas by Axiome is understood that which is claimed as due. So that between [...] and [...] there is this other difference, that this later is simply a Petition, the former a Petition of Right.

Of Petitions simply, the first is, That from any Point to any Point may be drawn a straight Line. The second, That a finite straight Line may be produced. The third, That upon any Center, at any distance may be described a Circle. All which are both evident, and necessary to be granted.

And by all these a man may easily perceive that Euclide in the definitions of a Point, a Line, and a Superficies, did not intend that a Point should be Nothing, or a Line be without Latitude, or a Superficies without Thickness, for if he did, his Petitions are not onely unreasonable to be granted, but also impossible to be performed. For Lines are not drawn but by Motion; and Motion is of Body only. And therefore his meaning was, that the Quantity of a Point, the Breadth of a Line, and the Thickness of a Superficies were not to be considered, that is to say, not to be reckoned in the demonstration of any Theoreme concerning the Quantity of Bodies, either in Length, Superficies, or Solid.

Of the Faults that Occurre in Demon­stration. To the same egregious Professors of the Ma­thematicks in the University of Oxford.

THere be but two causes from which can spring an error in the demonstration of any con­clusion in any Science whatsoever. And those are Ignorance or want of understanding, & Negligence. For as in the adding together of many and great Numbers, he cannot fail, that knoweth the Rules of Addition, and is also all the way so carefull, as not to mistake one number, or one place for another; so in any other Science, he that is perfect in the Rules of Logick, and is so watchfull over his Pen, as not to put one word for another, can never fail of making a true, though not perhaps the shortest and easiest demonstration.

The Rules of Demonstration are but of two kindes; One, that the Principles be true and evi­dent Definitions; the other, that the Inferences be necessary. And of true and evident defini­tions, the best are those which declare the cause or generation of that Subject, whereof the proper passions are to be demonstrated. For Science is that knowledge which is derived from the com­prehension of the cause. But when the cause appeareth not, then may, or rather must we define some known property of the Subject, and from thence derive some possible way, or wayes of the [Page 10] generation. And the more wai [...] of generation are explicated, the more easie will be the derivati­on of the Properties; whereof some are more immediate to one, some to another generation. He therefore that proceedeth from untrue, or not unde [...]d definitio [...], is ignorant of that he goes about; which is an il-favoured fault, be the matter he undertaketh easie or difficult; because he was not forced to undergo a greater charge then he could carry through. But he that from right definitions maketh a false conclusion, erreth through humane frailty, as being less awake, more troubled with other thoughts, or more in haste when he was in writing. Such faults, unless they be very frequent, are not attended with shame, as being common to all men, or are at least less ug­ly then the former, except then, when he that committeth them reprehendeth the same in other men. For that is in every man intolerable, which he cannot tolerate in another. But to the end that the faults of both kinds may by every man be well understood, it will not be amiss to examine them by some such Demonstrations, as are publikely extant. And for this purpose I will take such as are in mine and in your Books, and begin with your Elenchus of the Geome­try contained in my Book de Corpore; to which I will also joyn your Book lately set forth con­cerning the Angle of Contact, Conique sections, and your Arithm tica Infinitorum; and then examine the rest of my Philosophy, and yours that oppugne it. For I will take leave to consider you both every where, as one Author, because you publikely declare your approbation of one anothers doctrine.

My first Definition is of a Line, of Length, and of a Point; The way (say I) of a Body moved, in which magnitude (though it alwayes have some magnitude) is not considered, is called a Line; and the space gone over by that motion, Length, or one and a simple Dimension. To this Definition you say, First, what Mathematician did ever thus define a Line or Length? Whether you call in others for help or testimony, it is not done like a Geometrician; for they use not to prove their conclusions by witnesses, but relye upon the strength of their own reason; and when your witnesses appear, they will not take your part. Secondly, you grant that what I say is true, but not a Definition. But to tell you truly what it is which we call a Line, is to de­fine a Line. Why then is not this a Definition? Because (say you in the first place) it is not a reciprocall proposition. But by your favour it is reciprocall. For not only the way of a Bo­dy whose Quantity is not considered, is a Line; but also every Line is (or may be conceived to be) the way of a Body so moved. And if you object that there is a difference between is and may be conceived to be, Euclide whom you call to your aid, will be against you in the fourteenth Definition before his eleventh Element; Where he defines a Sphere just as convertibly as I de­fine a Line; except you think the Globes of the Sun and Stars cannot be Globes, unless they were made by the circumduction of a Semicircle; And again in the eighteenth definition, which is of a Cone, unless you admit no Figure for a Cone, which is not generated by the Revolu­tion of a Triangle; And again, in the twentieth Definition, which is of a Cylinder, except it be generated by the circumvolution of a Parallelogram. Euclide saw that what proper pas­sion soever should be derived from these his Definitions, would be true of any other Cylinder, Sphere, or Cone, though it were otherwise generated; And the description of the generation of any one being by the imagination applicable to all (which is equivalent to convertible) he did not believe that any rationall man could be missed by learning Logick, to be offended with it. Therefore this exception proceedeth from want of understanding, that is, from ignorance of the nature, and use of a Definition.

Again, you object and ask What need is there of motion, or [...]f Body moved, to make a man understand what is a Line? Are not Lines in a Body at rest, as well as in a Body moved? And is not the distance of two resting points Length, as well as the measure of the passage? Is not Length one and a simple dimension, and one and a simple dimension Line? Why then is not Line and Length all one? See how impertinent these questions are. Euclide defines a Sphere to be a Solid Figure described by the revolution of a Semicircle, about the unmoved Diameter. Why do you not ask what need there is to the understanding of what a Sphere is, to bring in the motion of a Semicircle? Is not a Sphere to be understood without such motion? Is not [Page 11] the Figure so made, a Sphere without this motion? And where he defines the Axis of a Sphere to be that unmoved Diameter, may not you ask, whether there be no Axis of a Sphere, when the whole Sphere, Diameter and all is in motion? But it is not my purpose to defend my definition by the example that of Euclide. Therefore first, I say, to me howsoever it may be to others, it was fit to define a Line by Motion. For the generation of a Line is the Motion that describes it. And having defined Philosophy in the beginning, to be the knowledge of the properties from the generation, it was fit to define it by its generation. And to your question, Is not distance Length? I answer, that though sometimes distance be aequivalent to Length, yet certainly the distance between the two ends of a thread wound up into a Clew is not the length of the thread; for the length of the thread is equall to all the windings whereof the Clew is made. But if you will needs have distance and length to be all one, tell me of what, the di­stance between any two Points is the Length. Is it not the length of the way? And how is that called Way, which is not defined by some motion? And have not severall wayes between the same places, as by Land and by Water, severall lengths? But they have but one distance, be­cause the distance is the shortest way. Therefore between the length of the Path, and the di­stance of the Places, there is a reall difference in this case, and in all cases a difference of the consideration. Your objection, that Line is Longitude, proceeds from want of understan­ding English. Do men ever ask what is the Line of a thread, or the Line of a Table, or of any other Body? Do they not alwayes ask what is the length of it? And why, but because they use their own judgements, nor yet corrupted by the subtilty of mistaken Professors. Euclide defines a Line, to be length without breadth. If those terms be all one, why said he not that a Line, is a Line without breadth? But what Definition of a Line give you? None. Be contented then with such as you receive, and with this of mine, which you shall presently see is not amiss.

Your next objections are to my Definition of a Point. Which Definition adhereth to the former in these words, and the Body it self is called a Point. Here again you call for help; Quis unquum mortalium, &c. What mortall man, what sober man did ever so Define a Point? 'Tis well, and I take it to be an honour to be the first that do so. But what objection do you bring against it? This, that a point added to a Point (if it have magnitude) makes it greater. I say it doth so, but then presently it loseth the name of a Point, which name was gi­ven to signifie that it was not the meaning of him that us'd it in demonstration, to add, substract, multiply, divide, or any way compute it. Then you come in with perhaps you will say though it have magnitude, that magditude is not considered. You need not say perhaps. You know I affirm it; and therefore your Argument might have been left out, but that it gave you oc­casion of a digression into scurvie language.

And whereas you ask why I defined not a Point thus, Punctum est corpus quod non conside­ratur esse corpus, & magnum quod non consideratur esse magnum. I will tell you why. First, because it is not Latine. Secondly, because when I had defined it by corpus, there was no need to Define it again by magnum. I understand very well this language, Punctum est corpus, quod non consideratur ut corpus. A Point is a Body not considered as Body. But Punctum, est cor­pus, quod non consideratur esse corpus, vel esse magnum, is not Latine, nor the version of it, A Point is a Body which is not considered to be a Body, English. My Definition was, that a Point is that Body whose magnitude is not considered, not reckoned, not put to account in Demonstration. And I exemplified the same by the Body of the Earth describing the Eclipcique Line; because the magnitude is not there reckoned nor chargeth the Eccliptique Line with any breadth. But I perceive you understand not what the word consideration signifieth, but take it for comparison or relation; and say I ought to define a Point simply, and not by relation to a greater Body; as if to reckon, and to compare were the same thing. Omnia mihi (saith Ci­cero) provisa & cousiderata sunt. I have provided and reckoned every thing. There is a great difference between Reckoning and Relation.

Again, you ask why Corpus motum, a Body moved. Ile tell you; because the [...] was [Page 12] necessary for the generation of a Line. And though after the generation of the Line, the Point should rest, yet it is not necessary from this Definition that it should be no more a Point; nor when Euclide defines a Sphere by the circumduction of a Semicircle upon an Axis that resteth, doth it follow from thence when the Sphere, Axis, Center and all (as that of the Earth) is moved from place to place, that it is no more an Axis.

Lastly, you object that motion is accidentary to a Point, and consequently not essentiall, nor to be put into the Definition. And is not the circumduction of a Semicircle accidentary to a Sphere? Or do you think the Sphere of the Sun was generated by the revolution of a Semi­circle? And yet it was thought no fault in Euclide to put that motion into the Definition of a Sphere.

The conceit, you have concerning Definitions, that they must explicate the essence of the thing defined, and must consist of a genus, and a difference, is not so universally true as you are made believe, or else there be very many insufficient Definitions that pass for good with you in Euclide. You are much deceived if you think these wofull notions of yours, and the Lan­guage that doth every where accompany them, shew handsomely together. Or that such grounds as these be able to sustain so many, and so haughty reproaches as you advance upon them, so as they fall not (as you shall see immediately) upon your own head. I say a point hath Quantity, but not to be reckoned in Demonstrating the properties of Lines, Solids, or Superficies; You say it hath no Quantity at all, but is plainly Nothing.

The first of the Petitions of Euclide is, that a Line may be drawn from Point to Point at any distance. The second, that a straight Line may be produced. The third, that on any Center a circle may be described at any distance. And the eighth Axiome (which Sir H. Savile observes to be the foundation of all Geometry) is this, Quae sibi mutuo congruunt, &c. Those things that are applyed one to another in all Points, are equall. All or any of these Principles being taken away, there is not in Euclide one Proposition Demonstrated, or Demonstrable. If a Point have not Quantity, a Line can have no Latitude; and because a Line is not drawn but by motion, by motion of a Body, and Body imprinteth Latitude all the way, it is impossible to draw or produce a straight Line, or to describe a circular Line without Latitude. Also if a Line have no Latitude, one straight Line cannot be applyed to another. To them therefore that deny a Point to have Quantity, that is, a Line to have Latitude, the forenamed Principles are not possible, and consequently no proposition in Geometry is demonstrated, or demonstrable. You therefore that deny a Point to have Quantity, and a Line to have Breadth, have nothing at all of the Science of Geometry. The practise you may have; but so hath any man that hath learned the bare Propositions by heart; but they are not fit to be Professors either of Geometry, or of any other Science that dependeth on it. Some man perhaps may say that this controversie is not much worth, and that we both mean the same thing. But that man (though in other things prudent enough) knoweth little of Science and Demonstration. For Definitions are not onely used to give us the Notions of those things whose appellations are defined; for many times they that have no Science have the Ideas of things more perfect then such as are raised by Definitions. As who is there that understandeth not better what a straight Line is, or what Proportion is, and what many other things are without Definition, then some that set down the Definitions of them. But their use is, when they are truly and clearly made, to draw Arguments from them for the Conclusions to be proved. And therefore you that in your following censures of my Geometry, take your Argument so often from this, That a Point is nothing, and so often revile me for the contrary, are not to be allowed such an excuse. He that is here mistaken, is not to be called Negligent in his Expression, but Ignorant of the Science.

In the next place, you take exceptions to my Definition of Equall Bodies, which is this, Cor­pora aequalia sunt quae eundem locum possidere possunt. Equall bodies are those which may have the same place. To which you object impertinently, that I may as well define a man to be He that may be Prince of Transilvania, Wittily, as you count wit. Formerly in every Definition, you exacted an Explication of the Essence. You are therefore of opinion that the Possibility of be­ing [Page 13] Prince of Transilvania, is no less Essentiall to a man, then the Possibility of the being of two Bodies successively in the same place, is Essentiall to Bodies equall.

You take no notice of the twenty third Article of this same Chapter, where I define what it is we call Essence; namely, that Accident, for which we give the thing its name. As the Es­sence of a man is his Capacity of reasoning, the Essence of a white-body, whiteness, &c. be­cause we give the name of man to such Bodies as are capable of Reasoning, for that their capa­city; and the name of White to such Bodies as have that colour, for that colour. Let us now examine why it is that men say Bodies are one to another Equall; and thereby we shall be able to determine whether the possibility of having the same place, be Essentiall or not to Bodies equall, and consequently whether this Definition be so like to the Defining of a man by the Possibility of being Prince of Transilvania, as you say it is. There is no man (besides such Egregious Geometricians as your selves) that inquireth the equality of two bodies, but by measure. And for Liquid Bodies, or the Aggregates of innumerable small Bodies, men (men I say) measure them by putting them one after another into the same vessell, that is to say, into the same place (as Aristotle defines place) or into the space determined by the vessell, as I define place. And the Bodies that so fill the vessell, they acknowledge, and receive for equall. But though, when hard Bodies cannot be so measured, without the incommodity, or trouble of al­tering their Figure, they then enquire (if the Bodies are both of the same kind) their equa­lity by weight, knowing (without your teaching) that equall bodies of the same nature, weigh Proportionably to their magnitudes; yet they do it not for fear of missing of the equality, but to avoid inconvenience, or trouble But you, (you I say) that have no Definition of Equalls, neither received from others, nor framed by your selves, out of your shallow meditation, and deep conceit of your own Wits, contend against the common light of Nature. So much is un­heedy learning a hinderance to the knowledge of the truth, and changeth into Elves those that were beginning to be men.

Again, when men inquire the equality of two Bodies in length, they measure them by a com­mon measure; in which measure they consider neither breadth nor thickness, but how the length of it agreeth; first, with the length of one of the Bodies, then with the length of the other. And both the Bodies whose lengths are measured, are successively in the same place under their common measure. Place therefore in Lines also, is the proper Index and discoverer of Equality, and Inequality. And as in length, so it is in breadth and thickness, which are but Lengths otherwise taken in the same Solid Body. But now when we come from this Equa­lity, and Inequality of Lengths known by measure, to determine the Proportions of Superficies, and of Solids, by ratiocination, then it is that we enter into Geometry; for the making of De­finitions, in whatsoever Science they are to be used, is that which we call Philosophiaprima. It is not the work of a Geometrician, as a Geometrician, to Define what is Equality, or Propor­tion, or any other word he useth, though it be the work of the same man, as a man. His Geo­metricall part is, To draw from them, as many true and usefull Theoremes as he can.

You object secondly, That a Pyramis may be equall to a Cube, whilst it is a Pyramis. True, And so also whilst it is a Pyramis it hath a possibility by flexion and transposition of parts to become a Cube, and to be put into the place where another Cube equall to it was before. This is to argue like a child that hath not yet the perfect understanding of any Language.

In the third and fourth objection, you teach me to define Equall Bodies (it I will needs de­fine them by place) by the Equality of place, and to say that Bodies are Equall that have Equall places. Teach others, if you can, to measure their grain, not by the same, but Equall Bushels.

In the fifth objection, you except against the word can, in that I say that Bodies are Equall, which can fill the same place. For the greater Body can (you say) fill the place of the less, though not reciprocally the less of the greater; It is true, that though the place of the less, can never be the place of the greate r yet it may be filled by a part of the greater. But 'tis not then the grea­ter Body that filleth the place of the less, but a part of it, that is to say, a less Body. Howsoever [Page 14] to take away from simple men this straw they stumble at, I have now put the Definition of Equal Bodies into these words, Equall Bodies are those whereof every one can fill the place of every other. And if my Definition displease you, propound your own, either of Equall Bodies, or of Equals simply. But you have none. Take therefore this of mine.

The sixth is a very admirable exception. What (say you) if the same Body can sometimes take up a greater, sometimes a lesser place, as by Rarefaction and Condensation? I understand very well that Bodies may be somtimes thin and sometimes thick, as they chance to stand closer together, or further from one another. So in the Mathematick-Schools, when you read your Lear­ned Lectures, you have a thick or thronging Audience of Disciples, which in a great Church, would be but a very thin company. I understand how thick and thin may be attributed to bo­dies in the Plurall, as to a company, but I understand not how any one of them is thicker in the School, then in the Church; or how any one of them taketh up a greater room in the School (when he can get in) then in the street. For I conceive the Dimensions of the Body, and of the Place, whether the place be filled with Gold or with Air, to be coincident and the same; and conse­quently both the Quantity of the Air, and the Quantity of the Gold to be severally equall to the Quantity of the place; and therefore also (by the first Axiome of Euclide) equall to one an­other; insomuch, as if the same Air should be by Condensation contained in a part of the place it had, the dimensions of it would be the same with the dimensions of part of the place, that is, should be less then they were, and by consequence the Quantity less. And then either the same body must be less also, or we must make a difference between greater Bodies, and Bodies of greater Quantity; which no man doth that hath not lost his wits by trusting them with absurd teach­ers. When you receive Salary, if the Steward give you for every shilling a piece of six pence, and then say, every shilling is condensed into the room of a six pence, I believe you would like this Doctrine of yours much the worse. You see how by your ignorance you confound the af­fairs of mankind, as far forth as they give credit to your opinions, though it be but little. For nature abhorres even empty words, such as are (in the meaning you assign them) Rarefying and Condensing. And you would be as well understood if you should say (coining words by your own power) that the same Body might take up sometimes a greater, sometimes a lesser place, by Wal­lifaction and Wardensation, as by Rarefaction and Condensation. You see how admirable this your objection is.

In the seventh objection you bewray another kind of Ignorance, which is the Ignorance of what are the proper works of the severall parts of Philosophy. Though it were out of doubt (say you) that the same Body cannot have several Magnitudes yet seeing it is matter of Natural Phi­losophy, nor hath any thing to do with the present business, to what purpose is it to mention it in a Mathematicall Definition? It seems by this, that all this while you think it is a piece of the Geometry of Euclide, no less to make the Definitions he useth, then to infer from them the Theorems he demonstrateth. Which is not true. For he that telleth you in what sense you are to take the Appelations of those things which he nameth in his discourse, teacheth you but his Language, that afterwards he may teach you his Art. But teaching of Language is not Ma­thematick, nor Logick, nor Physick, nor any other Science; and therefore to call a Definition (as you do) Mathematicall, or Physicall, is a mark of Ignorance (in a Professor) unexcusable. All Doctrine begins at the understanding of words, and proceeds by Reasoning till it conclude in Science. He that will learn Geometry must understand the Termes before he begin, which that he may do, the Master demonstrateth nothing, but useth his Naturall prudence onely, as all men do, when they endeavour to make their meaning clearly known. For words understood are but the seed, and no part of the harvest of Philosophy. And this seed was it, which Aristotle went about to sow in his twelve Books of Metaphysicks, and in his eight Books concerning the Hearing of Naturall Philosophy. And in these Books he defineth Time, Place, Substance or Essence, Quantity, Relation, &c. that from thence might be taken the Definitions of the most generall words for Principles in the severall parts of Science. So that all Definitions proceed from common understanding; of which, it any man rightly write, he may properly call his [Page 15] writing Philosophia prima, that is, the Seeds, or the Grounds of Philosophy. And this is the Method I have used, defining Place, Magnitude, and the other the most generall Appellations in that part which I intitle Philosophia prima. But you now not understanding this, talk of Ma­thematicall Definitions. You will say perhaps that others do the same as well as you. It may be so. But the appeaching of others does not make your ignorance the less.

In the eighth place you object not, but ask me why I define equall Bodies apart. I will tell you. Because all other things which are said to be equall, are said to be so, from the equality of Bodies; as two lines are said to be equall, when they be coincident with the Length of one and the same Body; and equall Times, which are measured by equall Lengths of Body, by the same Motion. And the reason is, because there is no Subject of Quantity, or of Equality, or of any other accident but Body; all which I thought certainly was evident enough to any un­corrupted Judgement; and therefore, that I needed first to define Equality in the Subject there­of, which is Body, and then to declare in what sense it was attributed to Time, Motion, and other things that are not Body.

The ninth objection is an egregious cavill. Having set down the Definition of Equall Bo­dies, I considered that some men might not allow the attribute of Equality to any things, but those which are the Subjects of Quantity, because there is no Equality, but in respect of Quan­tity. And to speak rigidly, Magnum & Magnitudo are not the same thing; for that which is great, is properly a Body, whereof greatness is an Accident. In what sense therefore (might you object) can an Accident have Quantity? For their sakes therefore that have not Judge­ment enough to perceive in what sense men say the Length is so Long, or the Superficies so broad, &c. I added these words, Eâdem ratione (quâ scilicet corpora dicuntur aequalia) Mag­nitudo magnitudini aequalis dicitur, that is, in the same manner, as Bodies are said to be equal, their magnitudes also are said to be equall. Which is no more then to say, when Bodies are Equall, their Magnitudes also are called Equall. When Bodies are Equall in Length, their Lengths are also called Equall. And when Bodies are Equall in Superficies, their Superfi­cies are also called Equall. All which is common speech, as well amongst Mathematicians, as amongst common people; and (though improper) cannot be altered, nor needeth to be al­tered to intelligent men. Nevertheless I did think fit to put in that clause, that men might know what it is we call Equality, as well in Magnitudes as in Magnis, that is, in Bodies. Which you so interpret, as if it bore this sense, that when Bodies are Equall, their Superficies also must be Equall, contrary to your own knowledge, onely to take hold of a new occasion of revi­ling. How unhandsome, and unmanly this is, I leave to be judged by any Reader that hath had the fortune to see the world, and converse with honest men.

Against the fourteenth Article, where I prove that the same Body hath alwayes the same magnitude, you object nothing but this, that though it be granted, that the same Bo [...]y hath the same magnitude, while it resteth, yet I bring nothing to prove that when it changeth place, it may not also change its Magnitude by being enlarged or co [...]tracted. There is no doubt, but to a Body (whether at rest or in motion) [...]o [...]e Body may be added, or part of it taken away. But then it is not the same Body, unless the Whole, and the Part be all one. It the Schools had not set your wit awry, you could never have been so stupid as not to see the weak­ness of such objections. That which you add in the end of your objections to this eighth Chap­ter, that I allow not Euclide this Axiom gratis, that the Whole is greater then a Part, you know to be untrue

At my eleventh Chapter, you enter into dispute with me, about the nature of Proportion. Upon the truth of your Doctrine therein, and partly upon the truth of your opinions concerning the Definitions of a Point, and of a Line, dependeth the Question whether you have any Geo­metry, or none; and the truth of all the Demonstrations you have in your other Books, namely, of the Angle of Contact, and Arithmetica Infinitorum. Here I say you e [...]ter, how you wil get out (your reputation saved) we shall se [...] hereafter.

When a man asketh what Proportion one Quantity hath to another, he asketh how great or [Page 16] how little the one is comparatively to, or in respect of the other. When a G [...]ometrician prefixeth before his Demonstrations a D [...]inition, he doth it not as a part of his Geometry, but of natu­rall evidence, not to be demonstrated by Argument, but to be understood in understanding the Language▪ wherein it is set down; though the matter may nevertheless (if besides Geometry he have wit) be of some help to his Disciple to make him understand it the sooner. But when there is no [...]ignificant Definition prefixed (as in this case, where Euclides Definition of Propor­tion, That it is a whats [...]i [...]a [...]t habitu [...]e of two Quantities, &c. is in significant, and you al­ledge no other) every one that will learn Geometry, must gather the Definition from observing how the word to be defined is most constantly used in common speech. But in common speech if a man [...]hall ask how much (for example) is six in respect of four, and one man answer that it is greater by two, and another that it is greater by half of four, or by a third of six, he that asked the question, will be satisfied by one of them, though perhaps by one of them now, and by the other another time, as being the onely man that knoweth why he himself did ask the Question. But if a man should answer, as you would do, that the Proportion of six to two is of th [...]se numbers, a certain Quotient, he would receive but little satisfaction. Between the said answers to this Question, How much is six in respect of four? there is this difference. He that answereth that it is more by two, compareth not two with four, nor with six, for two is the name of a Quantity absolute. But he that answereth it is more by half of four, or by a third of six, compareth the difference with one of the differing Quantities. For halfs, and thirds, &c, are names of Quantity compared.

From hence there ariseth two Species or kinds of (Ratio) Proportion, into which the gene­rall word Proportion may be divided. The one whereof (namely, that wherein the Difference is not compared with either of the differing Quantities) is called (Ratio Aritbmetica) Arith­meticall Proportion; the other (Ratio Geometrica) Geometricall Proportion, and (because this latter is onely taken notice of by the name of Proportion) simply Propor­tion. Having considered this, I defined Proportion, Chap. 11. Arti. 3. in this manner, Ratio est Relatio Antecedentis ad Consequens secundum magnitudinem. Proportion is the Relation of the Antecedent to the Consequent in Magnitude; having immediately before defined Rela­tives, Antecedent, and Consequen [...] in the same Article, and by way of explication added, that such Relation was nothing else but that one of the Quantities was equall to the other, or ex­ceeded it by some Quantity, or was by some Quantity exceeded by it. And for exemplifica­tion of the same, I added further, that the Proportion of three to two, was that three exceeded two by a unity; but said not that the unity, or the difference whatsoever it were, was their pro­portion, for Unity, and to exceed another by Unity, is not the same thing. This is cleer enough to others. Let us therefore see why it is not so to you. You say I make Proportion to consist in that which remaineth after the lesser Quantity is substracted out of the greater; and that you make it to consist in the Quotient, when one number is divided by the other. Wherein you are mistaken; first, in that you say I make the Proportion to consist in the Remainder. For I make it to consist in the act of exceeding, or of being exceeded, or of being equall; whereas the Re­mainder is alwayes an absolute Quantity, and never a Proportion. To be more or less then ano­ther number by two, is not the number two; Likewise to be equall to two, where the difference is nothing, is not that nothing? Again, you mistake in saying the Proportion consisteth in the Quotient. For divide twenty by five, the Quotient is four. Is it not absurd to say that the Proportion of five to twenty, or of twenty to five, is four? You may say the Proportion of five to twenty, is the Proportion of one to four. And so say I. And you may therefore also say, that the Proportion of one to four is a measure of the Proportion of five to twenty, as being Equall. And so say I. But that is onely in Geometricall Proportion, and not in Proportion universal­ly. For though the Species obtain the Denomination of the Genus; yet it is not the Genus. But as the Quotient giveth us a measure of the Proportion of the Dividend to the Divisor in Geometricall Proportion, so also the Remainder after Substraction is the measure of Proportion Arithmeticall.

[Page 17] You object in the next place, That if the Proportion of one Quantity to another be nothing but the excess or defect, then, wheresoever the Excess or Defect is the same, there the Propor­tion is the same. This you say follows in your Logick, and from thence, that the Proportion of three to two, and five to four is the same. But is not three to two, and five to four, where the Excess is the same number, the same Proportion Arithmeticall? And is not Arithmeticall Proportion, Proportion? You take here (Ratio) Proportion (which is the Genus) for that Species of it, which is called Geometricall, because usually this Species has the name of Pro­portion simply. Also that the Proportion of three to two, is the same with that of nine to six; Is it not because the excesses are one and three, the same portions of three and nine, that is to say, the same excesses comparatively? I wonder you ask me not what is the G [...]nus of Arithme­ticall, and Geometricall Proportions; and what the Difference. The Genus is (Ratio) Pro­portion, or Comparison in Magnitude, and the Diff [...]rence is that one Comparison is made by the absolute Quantity, the other by the Comparative Quantity, of the Excess or Defect, if there be any. Can any thing be clearer then this? You a [...]ter come in with Ignosce hebitudi­ni to no purpose. I am not so inhumane as not to pardon dulness, or madness; They are not voluntary faults. But when men adventure voluntarily, to talk of that they understand not censoriously and scornfully, I may tell them of it.

This difference between the Excesses or Defects, as they are simply or comparatively reckoned, being thus explained, all the rest of that you say in your objections to this eleventh Chapter (sa­ving that Artt. 5. for Ratio binarii ad quinarium est superar [...] T [...]rnario, as it is in other places, I have put too hastily Ratio binarii ad quinarium est Ternarius) will be understood by every Reader to be frivolous, and to proceed from the ignorance of what Proportion is.

At the twelfth Chapter you onely note that I say, That the Pr [...]portion of Inequality is Quantity, but the Proportion of Equality not Quantity, and refer that which you have to say against it to the Chapter following; to which place, I shall also come in the following Lesson.

Of the Faults that Occurre in Demon­stration. To the same egregious Professors of the Ma­thematicks in the University of Oxford.

YOu begin your reprehension of my thirteenth Chapter with a Question. Whereas I divide Proportion into Arithmeticall, and Geometricall; You ask me what proportion it is I so divide. Euclide divides an Angle into Right, Obtuse, and Acute. I may ask you as pertinently what Angle it is he so divides? Or when you divice Animal into Homo, and Br [...]tum, what Animal that is which you so divide? You see by this how absurd your Q [...]estion is. But you say the Definition of Proportion which I make at Chap. 11. Art. 3. namely, that Pro­portion [Page 18] is the comparison of two Magnitudes, one to another, agreeth not, neither with Arith­meticall, nor with Geometricall Proportion. I believe you thought so then, but having read what I have said in the end of the last Lesson, if you think so still, your fault will be too great, to be pardoned easily. But why did you think so before? Is it not because there was no Definiti­on in Euclide of Proportion universall, and because he maketh no mention of Proportion Arith­meticall, and because you had not in your minds a sufficient notion thereof your selves to supply that Defect? And is not this the cause also, why you put in this Parenthesis (if Arithmeticall Proportion, ought to be called Proportion)? which is a confession that you know not whether there be such a thing as Arithmeticall Proportion or not; notwithstanding, that on all occasions, you speak of Arithmetically Proportionals. Yes, this was it that made you think that Propor­tion universally, and Proportion Geometricall is the same and yet to say you cannot tell whether they be the same or not. 'Tis no wonder therefore, if in such confusion of the understanding, you apprehend not that the Proportions of two to five, and nine to twelve, are the same; for you are blin­ded by seeing that they are not the same Proportions Geometricall. Nor doth it help you that I say the Difference is the Proportion, f [...] by Difference you might if you would, have understood the act of Differing.

At the second Article you note for a fault in Method, that after I had used the words Ant [...] ­cedent and Consequent of a Proportion in some of the precedent Chapters, I define them after­wards.. I do not believe you say this against your knowledge, but that the eagerness of your malice made you oversee. Therefore go back again to the third A [...]ticle of Chap. 11. Where ha­ving defined Correlatives, I add these words, Of which the first is called the Antecedent, the second the Consequent. This is but an oversight, though such as in me, you would not have excused.

At the thirteenth Article you find fault with, that I say that the Proportion of Inequality, whe­ther it be of Excess, or of Defect, is Quantity, but the Proportion of Equality is not Quan­ty. Whether that which you say, or that which I say be the truth, is a Question worthy of a very strict Examination. The first time I heard it argued, was in Mersennus his Chamber at Paris, at such time, as the first volume of his Cogitata Physico-Mathematica was almost prin­ted: In which, because he had not said all he would say of Proportion, he was forced to put the rest into a Generall Preface; which as was his custom, he did read to his Friends, before he sent it to the Pr [...]ss. In that generall Preface under the Title de Rationibus at (que) Proportionibus, at the Numbers twelve, thirteen, fourteen, he maintaineth against Clavius, that the Composition of Proportions is (as of all other things) a Composition of the Parts to make a Totall; and that the proportion of equality answereth in Quantity, to non-ens, or Nothing; the proportion of excess, to ens, or Quantity; and the Proportion of Defect, to less then Nothing; because Equality (he saies) is a term of middle signification, between Ex [...]ess and Defect. And there also he refuteth the Arguments which Clavius at the end of the nineth Element of Euclide bringeth to the contrary. And though this were approved by divers good Geometricians then present, and never gain-sayed by any since, Yet do not I say it upon the credit of them, but upon sufficient grounds. For it hath been demonstrated by Euto [...]ius that if there be three magnitudes, the proportion of the first to the third is compounded of the proportions of the first to the second, and of the second to the third; Which also I demonstrate in this Article. And if there were never so many magnitudes ranked, it might be likewise demonstrated, that the Proportion of the first to the last is compounded of the Proportions of the first to the second, and of the second to the third, and of the third to the fourth, and so on to the last. If there­fore we put in order any three numbers, whereof the two last be equall, as four, seven, seven, the Proportion of four the first, to seven the last, will be compounded of the Proportions of four the first, to seven the second, and of seven the second, to seven the third. Wherefore the Proportion of seven to seven (which is of equality) addeth nothing to the Proportion of four the first, to seven the second; and consequently the Proportion of seven to seven hath no Quantity. But that the Proportion of Inequality hath Quantity, I prove it fro [...] this, that one Inequality may be greater then another.

[Page 19] But for the clearing of this Doctrine (which Mersennus cals Intricate) of the composition of Proportions, I observed, first, that any two Quantities, being exposed to sense, their Pro­portion was also exposed; which is not Intricate. Again, I observed that if besides the two exposed Quantities, there were exposed a third, so as the first were the least, and the third the greatest, or the first the greatest, and the third the least, that not onely the Proportions of the first to the second, but also (because the Differences, and the Quantities proceed the same way) the Proportion of the first to the last is exposed by composition, or addition of the Differences; nor is there any intricacy in this. But when the first is less then the second, and the second greater then the third, or the first greater then the second, and the second less then the third, so that to make the first and second equall, if we use addition, we must to make the second and third equall use substraction, then comes in the intricacy, which cannot be extricated, but by such as know the truth of this Doctrine which I now delivered out of Mersennus: namely, That the Proportions of Excess, Equality, and Defect, are as Quantity, not-Quantity, no­thing want Quantity, or as Symbo lists mark them [...]. 0. 0-1 And upon this ground I thought depended the universall truth of this Proposition, that in any rank of Mag­nitudes of the same kind, the Proportion of the first to the last, was compounded of all the Pro­portions (in order) of the intermediate Quantities; the want of the proof thereof Sir Henry S [...]vile [...]als (Naevus) a mole in the Body of Geometry. This Proposition is demonstrated at the thirteenth Article of this Chapter.

But before we come thither, I must examine the Arguments you bring to confute this Pro­position, that the Proportion of Inequality is Quantity, of Equality not Quantity.

And first, you object that Equality and Inequality are in the same Predicament. A pretty Argument to flesh a young Scholar in the Logick School, that but now begins to learn the Pre­dicaments. But what do you mean by Aequale, and Inequale? Do you mean Corpus Aequale, and Corpus Inequale? They are both in the Predicament of Substance, neither of them in that of Quantity; Or do you mean Aequalitas, and Inaequalitas? They are both in the Predi­cament of Relation, neither of them in that of Quantity, and yet both Corpus, and Inaequa­litas, though neither of them be Quantity, may be Quanta, that is, both of them have Quantity, And when men say Body is Quantity, or Inequality is Quantity, they are no otherwise understood, then if they had said Corpus est tantum, and Inaequalitas tanta, not Tantitas; that is, Bodies and Inequalities are so much, not somuchness; and all intelligent men are contented with that expression, and your selves use it. And the Quantity of Inequality is in the Predicament of Quantity, because the measure of it is in that Line by which one Quantity exceeds the other. But when neither exceedeth other, then there is no Line of Ex­cess, or Defect by which the Equality can be measured, or said to be so much, or be called Quantity. Philosophy teacheth us how to range our words; but Aristotles ranging them in his Predicaments, doth not teach Philosophy; And therefore no Argument taken from thence, can become a Doctor, and a Professor of Geometry.

To prove that the Proportion of Inequality was Quantity, but the Proportion of Equali­ty, not Quantity, my Argument was this; That because one Inequality may be greater or less then another, but one Equality cannot be greater nor less then another, Therefore Ine­quality hath Quantity, or is Tanta, and Equality not. Here you come in again with your Predicaments, and object, that to be susceptible of magis and minus belongs not to quantity, but to Quality; but without any proof, as if you took it for an Axiome. But whether true or false, you understand not in what sense it is true or false. 'Tis true, that one Inequality is In­equality, as well as another; as one heat is heat as well as another; but not as great; Tam, but not Tantus. But so it is also in the Predicament of Quantity; one Line is as well a Line as another, but not so great. All degrees, intensions, and remissions of Quality, are greater or less Quantity of force, and measured by Lines, Superficies, or Solid Quantity, which are properly in the Predicament of Quantity. You see how wise a thing it is to argue from the Predicaments of Aristotle, which you understand not. And yet you pretend to be less addi­cted to the authority of Aristotle, now, then heretofore.

[Page 20] In the next place you say, I may as well conclude from the not susception of greater and less, that a Right Angle is not Quantity, but an Oblique one is. Very learnedly. As if to be greater or less could be attributed to a Quantity once determined. Number (that is number indefinitely taken) is susceptible of greater and less, because one number may be greater then another. And this is a good Argument to prove that Number is Quantity. And do you think the Argu­ment the worse for this, that one six cannot be greater then another six? After all these childish Arguments which you have hitherto urged, can you perswade any man, or your selves, that you are Logicians?

To the fifth and sixth Article you object, first, that if I had before sufficiently Defined (Ra­tio). Proportion, I needed not again define what is (eadem Ratio) the same-Proportion; and ask me whether when I have defined man, I use to define anew what is the same man? You think when you have the Definition of Homo, you have also the Definition of idem Homo, when 'tis harder to conceive what idem signifies, then what Homo. Besides, idem hath not the same signification alwayes, and with whatsoever it be joyned; it doth not signifie the same with Homo, that it doth with Ratio. For with Homo, it signifies the same individual man, but with Ratio it signifies a like, or an equall Proportion. And both (Ratio) Proportion, and (idem) the same, being defined, there will still be need of another Definition for (eadem Ratio) the same Proportion. And this is enough to defend both my self and Euclide, against this objection. For Euclide also after he had Defined (Ratio Proportion, and that su [...]ciently as he believed, yet he defines the same Proportion again apart. I know you did not mean in this place to object any thing against Euclide; but you saw not what you were doing. There is within you some spec [...]all cause of Intenebration, which you [...]ld do well to look to.

In the nex [...] p [...]ce you say, when I had defined A [...]thmeticall Proportions to be the same when the difference is the same; it was to be expected I should define Geometricall Proportions to be then the same, when the Antecedents are of their Consequents Totuple, or Tantuple, that is, equimultiple (for Tantuplum signifies nothing.) In plain words, you exp [...]cted, that as I defi­ned one by substraction, I should define the other by the Quotient in Division. But why should you expect a Definition of the same Proportion by the Quotient? Neither Reason nor the Authority of Euclide could move you to expect it. Or why should you say it was to be expected? But it seems you have the vanity to place the measure of truth in your own Learning. In Lines incommensurable there may be the same Proportion, when nevertheless there is no Quo­tient; for setting their Symboles one above another doth not make a Quotient; for Quotient there is none, but in aliquot parts. It is therefore impossible to define Proportion universally, by comparing Quotients. This incommensurability of Magnitudes was it that confounded Euclide in the framing of his Definition of Proportion at the fifth Element. For when he came to numbers, he defined the same Proportion irreprehensibly thus, Numbers are then Proportio­nal, when the first of the second, and the third of the fourth are equimultiple, or the same part, or the same parts; and yet there is in this Definition no mention at all of a Quotient. For though it be true that if in dividing two Numbers you make the same Quotient, the Dividends and the Divisors are Proportionall, yet that is not the Definition of the same Proportion, but a Theoreme Demonstrable from it. But this Definition Euclide could not accommodate to Pro­portion in Generall, because of incommensurability.

To supply this want, I thought it necessary to seek out some way, whereby the Proportion of two Lines, Commensurable, or Incommensurable, might be continued perpetually the same. And this I found might be done by the Proportion of two Lines described by some uniform mo­tion, as by an Efficient cause both of the said Lines, and also of their Proportions. Which mo­tions continuing, the Proportions must needs be all the way the same. And therefore I defined those Magnitudes to have the same Geometricall Proportion, when some cause producing in equall times, Equall Effects, [...]id determine both the Proportions. This you say needs an Oedipus to make it understood. You are (I see) no Oedipus; but I do not see any diff [...]ulty, neither in the Definition, nor in the Demonstration. That which you call perplexity in the [Page 21] Explication, is your prejudice, arising from the Symboles in your fancy. For men that pre­tend no less to naturall Phylosophy, then to Geometry, to find fault with bringing Motion and Time into a Definition, when there is no effect in nature, which is not produced in Time by Motion, is a shame. But you swim upon other mens bladders in the Superficies of Geometry, without being able to indure diving. Which is no fault of mine; and therefore I shall (without your leave) be bold to say, I am the first that hath made the grounds of Geometry firm and coherent. Whether I have added any thing to the Edifice or not, I leave to be judged by the Readers. You see, you that profess with the pricking of bladders the letting out of their vapour, how much you are deceived. You make them swell more then ever.

For the Corollaries that follow this sixth Article, you say they contain nothing new. Which is not true. For the nineth is new, and the Demonstrations of all the rest are new, being groun­ded upon a new Definition of Proportion, and the Corollaries themselves for want of a good De­finition of Proportion, were never before exactly Demonstrated. For the truth of the sixth Definition of the fifth Elem [...]t of Euclide cannot be known but by this Definition of mine, be­cause it requires a Triall in all numbers possible, that is to say, an infinite time of Triall, whether the quimultiples of the first and third, and of the second and fourth in all multiplications do together exceed, together come short, and are together equall; which Triall is impossible.

In objecting against the thirteenth and sixteenth Article, I observe that you bewray together, both the greatest Ignorance, and the greatest malice; and 'tis well, for they are sutable to one another, and fit for one and the same man. In the thirteenth Article my Proposition is this, If there be three Magnitudes that have Proportion one to another, the Proportions of the first to the second, and of the second to the third taken together (as one Proposition) are equall to the Proportion of the first to the third. This Demonstrated, there is taken away one of those Moles which Sir Henry Savile complaineth of in the Body of Geometry. Let us see now what you say both against the Enunciation, and against the Demonstration.

Against the Enunciation you object, that other men wo [...]ld say (not the Proportions of the first to the second, and of the second to the third, taken together, &c. but) the Proportion which is compounded of the Proportion of the first to the second, and of the second to the third, &c. Is not the compounding of any two things whatsoever the finding of the sum of them both, or the ta­king of them together as one totall. This is that absurdity of which Mersennus in the generall Preface to his Cogitata Physico-Mathematica hath convinced Clavius, who at the end of Eu­clides nineth Element denyeth the composition of Proportion to be a Composition of Parts to make a Totall; Which therefore he denyed, because he did not observe, that the Addition of a Proportion of defect, to a Proportion of Excess, was a Substraction of Magnitude; and because he understood not that to say, Composition is not the making a whole of Parts, was contradicti­on; which all, but too learned men would as soon as they heard [...]bho [...]re. Therefore in saying that other men would not speak in that manner, you say in effect they would speak absurdly. You do well to mark what other Geometricians say; but you would do better if you could by your own Meditation, upon the things themselves, examine the truth of what they say. But you have no minde (you say) to contend about the Phrase. Let us therefore see what it is you con­tend about.

The Proportion (you say) which is compounded of double and triple Proportion, is not (as I would have it) Quintuple, but Sextuple, as in these numbers, six, t [...]ree, one; where the Pro­portion of six to three is double, the Proportion of three to one tripl [...], and the Proportion of six to one sextuple, not quintuple. Tell me (egregsous Professors) how is six to three double Pro­portion? Is six to three the double of a number, or the double of some Proportion? All men know the number six is double to the number three, and the number three triple to an unity. But is the Question here of compounding numbers, or of compounding Proportions▪ Euclide at the last Proposition of his nineth Element sayes indeed, that these numbers, one, two, four, eight, are [...] in double Proportion, yet there is no m [...]n that under­stands it otherwise, then if he had said in Proportion of the single Quantity, to the double Quan­tity; [Page 22] and after the same Rate, if he had said three, nine, twenty-seven, &c. had been in triple Proportion, all men would have understood it, of the Proportion of any Quantity to its Triple. Your instance therefore of six, three, one, is here impertinent, there being in them no doubling, no tripling, nor sextupling of Proportions, but of numbers. You may observe also that Eu­clide never distinguisheth between double and duplicate, as you do. One word [...] serves him every where for either. Though I confess some curious Grammarians take [...] for duplicate in number, and [...] for double in Quantity; which will not serve your turn. Your Geometry is not your own, but you case your selves with Euclides; in which, as I have shewed you, there be some few great holes; and you by misunderstanding him, as in this place, have made them greater, Though the beasts that think your railing, [...]oaring, have for a time ad­mired you; yet now that through these holes of your case, I have shewed them your ears, they will be less affrighted. But to exemplifie the composition of Proportions, take these numbers, thirty-two, eight, one, and then you shall see that the Proportion of thirty-two to one is the sum of the Proportions of thirty-two to eight, and of eight to one. For the Proportion of thirty-two to eight is double the Proportion of thirty-two to six­teen. And the Proportion of eight to one, is triple the Proportion of thirty-two to sixteen, and the Proportion of thirty-two to one is Quintiple of thirty-two to sixteen. But double and triple added together maketh quintuple. What can be here denyed?

My Demonstration consisteth of three cases. The first is when both the Proportions are of defect, which is then, when the first Quantity is the least; as in these three Quantities, AB, AC, AD. The first case I demonstrated thus. [...] Let it be supposed that the point A were moved uniformly through the whole line AD. The Proportions therefore of AB to AC, and of AC to AD, are determined by the difference of the Times in which they are described. And the Proportion also of AB to AD, is that which is determined by the difference of the Times, in which they are described; but the difference of the Times in which AB and AC are described, together with the difference of the Times, wherein AC and AD are de­scribed, is the same with the difference of the Times, wherein are described AB and AD. The same cause therefore which determines both the Proportions of AB to AC, and of AC to AD, determines also the Proportion of AB to AD. Wherefore by the Definition of the same Proportion, Article six, the Proportion of AB to AC, together with the Proportion of AC to AD is the same with the Proportion of AB to AD.

Consider now your argumentation against it. Let there be taken (say you) between A and B the Point a, and then in your own words, I argue thus. The difference of the Times wherein are described AB and AC, together with the difference of the Times, wherein are described AC and AD, is the same with the difference of the Times, in which are described aB and aC (namely BD, or BC † CD) wherefore the same cause which determines the two Proportions of AB to AC, and of AC to AD, determines also the Proportion of aB to aD. Let me ask you here whether you suppose the Motion from a to B, or from a to D, to have the same switfness with the motion from A to B, or from A to D. If you do not, then you de­ny the supposition. If you do, then BC which is the difference of the Times AB and AC, cannot be the difference of the Times in which are described aB and aC, except AB and aB are equall. Let any man judge now whether there be any Paralogism in Orontius that can equall this. And whether all that follows in the rest of this, and the next two whole Pages, be not all a kind of raving upon the ignorance of what is the meaning of Proportion, which you also make more ill-favoured by writing it; not in language, but in Gamboles, I mean in the Sym­boles, which have made you call those demonstrations short, which put into words so many as a true demonstration requires, would be longer then any of those of Clavius upon the twelfth Element of Euclide.

To the sixteenth Article you bring no Argument, but fall into a loud Oncethmus (the special Figure wherewith you grace you Oratory) offended with my unexpected crossing of the Do­ctrine [Page 23] you teach, that Proportion consisteth it a Quotient. For that being denyed you, your [...] comes to nothing, that is, to just as much as it is worth But are not you very simple men, to say that all Mathematicians speak so, when it is not speak­ing? When did you see any man but your selves publish his Demonstrations by signs not ge­nerally received, except it were not with intention to demonstrate, but to t [...]ch the use of Signes? Had Pappus no Analytiques? Or wanted he the wit to [...]ten his reckoning by Signes? Or has he not proceeded Analytically in an hundred Problems (especially in his seventh Book) and never used Symboles? Symboles are poor unhandsome (though necessary) sc [...]ffolds of Demonstration; and ought no more to appear in publique, then the most d [...]rmed neces­sary business which you do in your Chambers. But why (say you) is this [...]tion to the Proportion of the greater to the less? Ile tell you; because i [...]erating of the Proportion of the less to the greater, is a making of the Proportion less, and the defect greater. And it is absurd to say that the taking of the same Quantity twice should make it less. And thence it is, that in Quantities, which begin with the less, as one, two, four, the Proportion of one to two, is greater then that of one to four, as is Demonstrated by Euclide Elem. 5. Prop. 8. and by con­sequent the Proportion of one to four, is a Proportion of greater littleness then that of one to two. And who is there, that when he knoweth that the respective greatness of four to one, is double to that of the respective greatness of four to two, or of two to one, will not presently acknowledge that the respective greatness of one to two, or two to four is double to the respective greatness of one to four. But this was too deep for such men as take their opinions not from weighing, but from reading.

Lastly, you object against the Corollarie of Art. 28. (which you make absurd enough by re­hearsing it thus) si quantitas aliqua divisa supponatur in partes aliquot aequales numero infi­nitas. &c. Do you think that of partes aliquot, or of partes aliquotae, it can be said without absurdity, that they are numero infinitae? And then you say I seem to mean, that if of the Quan­tity AB, there be supposed a part CB, infinitely little, and that between AC and AB be taken two means, one Arithmeticall AE, the other Geometricall AD, the difference between AD, and AE, will be infinitely little. My meaningis, and is sufficiently expressed, that the said means taken every where (not in one place onely) will be the same throughout. And you that say there needed not so much pains to prove it, and think you do it shorter, prove it not at all. For why may not I pretend against your demonstration, that BE the Arithme­ticall difference, is greater then BD the Geometricall difference. You bring nothing to prove it, and if you suppose it, you suppose the thing you are to prove. Hitherto you have procee­ded in such manner with your Elenchus, as that so many objections as you have made, so ma­ny false Propositions you have advanced. Which is a peculiar excellence of yours, that for so great a stipend as you receive, you will give place to no man living for the number and grossness of errors you teach your Scholars.

At the fourteenth Chapter your first exception is to the second Article; where I define a plain in this manner. A plain Superficies is that which is described by a straight Line so moved, as that every Point thereof describe a severall straight L [...]. In which you require, first, that instead of describe, I should have said can describe. Why do you not require of Euclide in the Definition of a Cone, instead of (Continetur) is contained, he say (contineri potest) can be contained? It I tell you how one Plain is generated, cannot you apply the same generation to any other Plain? But you object that the Plain of a Circle may be generated by the motion of the Radius, whose every point describeth not a straight but a crooked Line, wherein you are deceived; for you cannot draw a Circle (though you can draw the perimeter of a Circle) but in a Plain already generated. For the motion of a straight Line, whose one Point resting, describeth with the other Points severall perimeters of Circles, may as well describe a Conique Superficies, as a Plain. The Question therefore is, how you will in your Definition take in the Plain which must be generated before you begin to describe your Circle, and before you know what Point to make your Center. This objection therefore is to no purpose; and besides, that [Page 24] it reflecteth upon the perfect definitions of Euclide before the eleventh Element; it cannot make good his Definition (which is nothing worth) of a Plain Superficies, before his first Ele­ment.

In the next place you reprehend briefly this Corollarie, that two Plaines cannot inclose a Solid. I should indeed have added, with the base on whose extreams they insist. But this is not a fault to be ashamed of. For any man by his own understanding might have mended my expression without departing from my meaning. But from your Doctrine that a Superficies has no thickness, 'tis impossible to include a Solid, with any Number of Plains whatsoever; un­less you say that Solid is included which nothing at all includes.

At the third Article, where I say of crooked lines, some are every where crooked, and some have parts not crooked. You ask me what crooked Line has parts not crooked; and I answer, it is that Line which with a straight Line makes a rectilineall Triangle. But this you say cannot stand with what I said before, namely, that a straight and crooked line cannot be coincident; which is true, nor is there any contradiction; for that part of a crooked line which is straight, may with a straight line be coincident.

To the fourth Article, where I define the Center of a Circle to be that Point of the Radius, which in the description of the Circle is unmoved; You object as a contradiction, that I had before defined a Point to be the body which is moved in the description of a Line. Foolishly, As I have already shown at your objection to Chap. 8. Art. 12.

But at the sixth Article, where I say that crooked, and incongruous Lines touch one another but in one Point, you make a cavill from this, that a Circle may touch a Parabola in two Points. Tell me truely, did you read and understand these words that followed, a crooked Line cannot be congruent with a straight line, because if it could, one and the same line should be both straight and crooked? If you did, you could not but understand the sense of my words to be this; when two crooked lines which are incongruous, or a crooked and a straight line touch one an­other, the contact is not in a Line, but only in one Point; and then your instance of a Circle and a Parabola, was a wilfull cavill, not befitting a Doctor. If you either read them not, or un­stood them not, it is your own fault. In the rest that followeth upon this Article, with your Diagram, there is nothing against me, nor any thing of use, novelty, subtilty or learning.

At the seventh Article, where I define both an Angle, simply so called, and an Angle of Con­tingence, by their severall generations, namely, that the former is generated when two straight Lines are coincident, and one of them is moved, and distracted from the other by circular mo­tion upon one common Point resting, &c. You ask me to which of these kinds of Angle, I ref [...]r the Angle made by a straight Line when it cuts a crooked Line. I answer easily and truly, to that kind of Angle which is called simply an Angle. This you understand not. For how (will you say) can that Angle which is generated by the divergence of two straight Lines, be other then Rectilineall? O, how can that Angle which is not comprehended by two straight Lines, be other then Curvilineall? I see what it is that troubles you, namely, the same which made you say before, that if the Body which describes a Line be a Point, then there is nothing which is not moved that can be called a Point. So you say here, If an Angle be generated by the motion of a straight Line, then no Angle so generated can be Curvilineall. Which is as well argued, as if a man should say, the House was built by the carriage and motion of Stone and Timber, there­fore when the carriage and that motion is ended, it is no more a house. Rectilineall and Curvi­lineall hath nothing to do with the nature of an Angle simply so called, though it be essentiall to an Angle of Contact. The measure of an Angle simply so called is a circumference of a Cir­cle, and the measure is alwayes the same kind of Quantity with the thing measured. The Re­ctitude or Curvity of the Lines which drawn from the Center intercept the Arch, is accidenta­ry to the Angle, which is the same, whether it be drawn by the motion circular of a streight line or of a crooked. The Diameter and the Circumference of a Circle make a right Angle, and the same which is made by the Diameter and the Tangent. And because the point of Contact is not (as you think) nothing, but a line unreckoned, and common both to the Tangent, and the [Page 25] Circumference, the same Angle computed in the Tangent is Rectilineall, but computed in the Circumference, not Rectilineall, but mixt; or, if two Circles cut one another, Curvi­lineall. For every Chord maketh the same Angle with the Circumference which it maketh with the line that toucheth the Circumference at the end of the Chord. And therefore when I divide an Angle simply so called into Rectilineall, and Curvilineall, I respect no more the ge­neration of it, then when I divide it into Right and Oblique. I then respect the generation, when I divide an Angle into an Angle simply so called, and an Angle of Contact. This that I have now said, if the Reader remember when he reads your objections to this, and to the nineth Article, he will need no more to make him see that you are utterly ignorant of the na­ture of an Angle, and that if ignorance be madness, not I, but you are mad; and when an An­gle is comprehended between a straight and a crooked Line (if I may compute the same Angle as comprehended between the same straight Line and the Point of Contact) that it is conso­nant to my definition of a Point by a Magnitude not considered. But when you in your trea­tise de Angulo Contactûs Chap. 3. Pag. 6. Lin. 8. have these words, Though the whole concur­rent Lines incline to one another, yet they form no Angle any where but in the very point of concourse, You, that deny a Point to be any thing, tell me how two nothings can form an Angle; or if the Angle be not formed neither before the concurrent Lines meet, not in the Point of concourse, how can you apprehend that any Angle can possibly be framed. But I wonder not at this absurdity, because this whole treatise of yours is but one absurdity continu­ed from the beginning to the end; as shall then appear when I come to answer your objections to that which I have briefly and fully said of that Subject in my 14. Chapter.

At the twelfth Article I confess your exception to my universall definition of Parallels to be just, though insolently set down. For it is no fault of ignorance (though it also infect the de­monstration next it) but of too much security. The Definition is this: Parallels are those Lines or Superficies, upon which two straight Lines falling, and wheresoever they fall, making equall Angles with them both, are equall; which is not, as it stands, universally true. But inserting these words the same way, and making it stand thus, Parallel Lines or Superficies are those, upon which two straight Lines falling the same way, and wheresoever they fall, making equall Angles, are equall, it is both true and universall; and the following Conse­ctary with very little change, as you may see in the translation, perspicuously demonstrated. The same fault occurreth once or twice more; and you triumph unreasonably, as if you had given therein a very great proof of your Geometry.

The same was observed also upon this place by one of the prime Geometricians of Paris, and noted in a Letter to his friend in these words, Chap. 14. Art. 12. the Definition of Paral­lels wanteth somewhat to be supplyed. And of the Consectary, he says, it concludeth not, because it is grounded on the Definition of Parallels. Truely, and severely enough, though without any such words as savour of Arrogance, or of Malice, or of the Clown.

At the thirteenth Article you recite the Demonstration by which I prove the Perimeters of two Circles to be Proportionall to their Semidiameters; and with Esto, fortasse, recte, omni­no, noddying to the severall parts thereof, you come at Length to my last inference; There­fore by (Chap. 13. Art. 6.) the Perimeters and Semidiameters of Circles are Proportionall; which you deny; and therefore deny, because you say it followeth by the same Ratiocination, that Circles also and Spheres are Proportionall to their Semidiameters. For the same distance (you say) of the Perimeter from the Center which determines the magnitude of the Semidia­meter, determines also the magnitude both of the Circle, and of the Sphere. You acknow­ledge that Perimeters and Semidiameters have the cause of their determination such as in equal times make equall spaces. Suppose now a Sphere generated by the Semidiameters, whilst the Semicircle is turned about. There is but one Radius of an infinite number of Radii, which describes a great Circle, all the rest describe lesser Circles Parallel to it, in one and the same time of Revolution. Would you have men believe, that describing greater and lesser Circles, is ac­cording to the supposition (temporibus aequalibus aequalia facere) to make equall spaces in equall [Page 26] times? Or when by the turning about of the Semidiameter is described the Plain of a Circle does it (think you) in equall times make the Plains of the interior Circles equall to the plains of the exterior? Or is the Radius that describes the inner Circles equall to the Radius that de­scribes the exterior? It does not therefore follow from any thing I have said in this demonstra­tion, that either Spheres, or Plains of Circles, are Proportionall to their Radii. And conse­quently all that you have said, triumphing in your own Incapacity, is said imprudently by your selves to your own disgrace. They that have applauded you, have reason by this time to doubt of all the rest that follows, and if they can, to dissemble the opinion they had before of your Geo­metry. But they shall see before I have done, that not only your whole Elenchus, but also your other Books of the Angle of Contact, &c. are meer ignorance and gibberish.

To the fourteenth Article you object, that (in the sixth figure) I assume gratis, that FG, DE, BC, are Proportionall to AF, AD, AB; and you referre it to be judged by the Reader. And to the Reader I referre it also. The not exact drawing of the Figure (which is now amended) is it that deceived you. For AF, FD, DB, are equall by construction. Also AG, GE, EC, are equall by construction. And FG, DK, BH, KE, HI, IC, are equall by Parallelism. And because AF, FG, are as the velocities wherewith they are described; also 2 AF (that is AD) and 2 FG (that is DE) are as the same veloci­ties. And finally 3 AF (that is AB) and 3 FG (that is BC) are as the same veloci­ties. It is not therefore assumed gratis, that FG, DE, BC, are Proportionall to AF, AD, AB, but grounded upon the sixth Article of the thirteenth Chapter; and consequently your objection is nothing worth. You might better have excepted to the placing of DE, first at adventure, and then making AD, two thirds of AB; for that was a fault, though not great enough to trouble a Candid Reader; yet great enough, to be a ground, to a malicious Reader, of a Cavill.

That which you object to the third Corollarie of Art. 15. was certainly a dream. There is no a [...]ing of an Angle CDE, for an Angle HDE, or BDE, neither in the Demonstra­tion, nor in any of the Corollaries. It may be you dream't of somewhat in the twentieth Ar­ticle of Chap. 16. But because that Article though once printed, was afterwards left out, as not serving to the use I had designed it for, I cannot guess what it is. For I have no Copy of that Article, neither printed nor written, but am very sure, though it were not usefull, it was true.

Article the sixteenth. Here we come to the Controversie concerning the Angle of Contact, which (you say) you have handled, in a speciall Treatise published; and that you have clearly demonstrated in your publick Lectures, that Peletarius was in the right. But that I agree not sufficiently neither with Peletarius, nor with Clavius. I confess I agree not in all points with Peletarius, nor in all points with Clavius. It does not thence follow that I agree not with the Truth. I am not (as you) of any faction, neither in Geometry, nor in Politicks. If I think that you, or Peletarius, or Clavius, or Euclide have e [...]red, or been too obscure; I see no cause, for which I ought to dissemble it. And in this same Question, I am of opinion that Peletarius did not well in denying the Angle of Contingence to be an Angle. And that Clavius did not well to say the Angle of a Semicircle was less then a Right-lined Right An­gle. And that Euclide did not well to leave it so obscure what he meant by Inclination in the Definition of a Plain Angle, seeing else where he attributeth Inclination onely to Acute An­gles, and scarce any man ever acknowledged Inclination in a straight Line, to any other Line, to which it was perpendicular. But you in this Question of what is Inclination, though you pretend not to depart from Euclide, are nevertheless more obscure then he; and also are con­trary to him. For Euclide by Inclination meaneth the Inclination of one Line to another; and you understand it of the Inclination of one Line from another, which is not Inclination, but Declination. For you make two straight. Lines when they lye one on another, to lye [...] that is without any Inclination (because it serves your turn); not observing that it followeth thence that Inclination is a digression of one Line from another. This is in your first [Page 27] Argument in the behalf of Peletarius (Pag. 10. Lin. 22.) and destroyshis opinion. For according to Euclide the greatest Angle is the greatest Inclination, and an Angle equall to two Right An­gles by this [...] should not be the greatest Inclination, as it is, but the least that can be. But if by the Inclination of two Lines we understand that proceeding of them to a common Point which is caused by their generation, which (I believ [...]) was Euclides meaning; then will the Angle of Contact be no less an Angle then a rectilineall Angle, but onely (as Clavius truly saies it is) Heterogeneous to it; and the doctrine of Clavius more conformable to Euclide then that of Peletarius. Besides, if it be granted you, that there is no inclination of the Circumfe­rence to the Tangent, yet it does not follow that their concourse doth not form some kind of Angle. For Euclide defineth there but one of the kinds of a Plain Angle. And then you may as much in vain seek for the Proportion of such an Angle to the Angle of Contact, as seek for the Focus, or Parameter of the Parabola of Dives and Lazarus. Your first argument there­fore is nothing worth, except you make good that which in your second Argument you affirm, namely, That all Plain Angles, not excepting the Angle of Contact, are (Homogeneous) of the same kind. You prove it well enough of other Curvilineall Angles; but when you should prove the same of an Angle of Contact, you have nothing to say but Pag. 17. Lin. 15. Unde autem illa quam somniet Heterogenia oriatur, neque potest ille ullatenus ostendere, neque ego vel somniare; whence should arise that diversity of kind, which he dreams of, neither can be at all shew, nor I dream; as if you knew what he could do if he were to answer you; or all were false which you cannot dream of. So that besides your customary vanity, here is nothing hitherto proved neither for the opinion of Peletarius, nor against that of Clavius. I have I think sufficiently explicated in the first Lesson, That the Angle of Contact is Quantity, name­ly, that it is the Quantity of that crookedness or flexion, by which a straight Line is bent into an Arch of a Circle equall to it; and that because the crookedness of one Arch may be grea­ter then the crookedness of another Arch of another Circle equall to it, therefore the Questi­on Quanta est curvitas, How much is the crookedness, is pertinent, and to be answered by Quantity. And I have also shewn you in the same Lesson, that the Quantity of one Angle of Contact is compared with that of another Angle of Contact, by a Line drawn from the Point of Contact, and intercepted by their Circumferences; and that it cannot be compared by any measure with a Rectilineall Angle.

But let us see how you answer to that which Clavius has objected already. They are Hetero­geneous, sayes he, because the Angle of Contact, how oft soever multiplyed, can never ex­ceed a Rectilineall Angle. To answer which you alleadge, it is no Angle at all; and that therefore it is no Angle at all because the Lines have no Inclination one to another. How can Lines that have no Inclination one to another, ever come together? But you answer, at least they have no Inclination in the Point of Contact. And why have two straight Lines In­clination before they come to touch, more then a straight Line and an Arch of a Circle? And in the Point of Contact it self, how can it be that there is less Inclination of the two Points of a straight Line and an Arch of a Circle, then of the Points of two straight Lines? But the straight Lines you say will cut; Which is nothing to the Question; and yet this also is not so evident, but that it may receive an objection. Suppose two Circles AGB and CFB to touch in B, and have a common Tangent through B. Is not the Line [figure] CFBGA a crooked Line? And is it not cut by the common Tan­gent DBE? What is the Quantity of the two Angles FBE and GBD, seeing you say neither DBG nor EBF is an Angle? 'Tis not therefore the cutting of a crooked Line, and the touching of it, that distinguisheth an Angle simply, from an Angle of Contact. That which makes them differ, and in kind, is, that the one is the Quantity of a Revolution, and the other the Quantity of Flexion.

In the seventh Chapter of the same Treatise, you think you prove the Angle of Contact, if it be an Angle, and a Rectilineall Angle to [Page 28] be (Homogeneous) of the same kind; when you prove nothing but that you understand not what you say. Those Quantities which can be added together, or substracted one from ano­ther, are of the same kind; But an Angle of Contact may be substracted from a right Angle, and the Remainder will be the Angle of a Semicircle, &c. So you say, but prove it not, unless you think a man must grant you that the Superficies contained between the Tangent and the Arch, which is it you substract, is the Angle of Contact; and that the Plain of the Semi­circle is the Angle of the Semicircle, which is absurd; though as absurd as it is, you say it di­rectly in your Elenchus, Pag. 35. Lin. 14. in these words, When Euclide defines a Plain An­gle to be the Inclination of two Lines, he meaneth not their aggregate, but that which lyes between them. It is true, he meaneth not the aggregate of the two Lines; but that he means that which lyes between them, which is nothing else but an indeterminate Superficies, is false, or Euclide was as foolish a Geometrician as either of you two.

Again, you would prove the Angle of Contact, if it be an Angle, to be of the same kind with a Rectilineall Angle, out of Eucl. 3. 16. Where he saies, it is less then any acute An­gle. And it follows well, that if it be an Angle, and less then any Rectilineall Angle; it is also of the same kind with it. But to my understanding Euclide meant no more, but that it was nei­ther greater nor equall; which is as truly said of Heterogeneous, as of Homogeneous Quan­tities. If he meant otherwise, he confirms the opinion of Clavius against you, or makes the Quantity of an Angle to be a Superficies, and indefinite. But I wonder how you dare venter to determine whether two Quantities be Homogeneous or not, without some Definition of Ho­mogeneous (which is a hard word) that men may understand what it meaneth.

In your eighth Chapter you have nothing but Sir H. Saviles Authority, who had not then resolved what to hold; but esteeming the Angle of contact, first, as others falsely did, by the Superficies that lyes between the Tangent and the Arch, makes the Angle of Contact, and a Rectilineall Angle Homogeneous; and afterwards, because no multiplication of the Angle of Contact can make it equall to the least Rectilineall Angle, with great ingenuity returneth to his former uncertainty.

In your nineth and tenth Chapters you prove with much ado, that the Angles of like Seg­ments are equall; as if that might not have been taken gratis by Peletarius without Demon­stration. And yet your Argument contained in the nineth Chapter is not a Demonstration, but a conjecturall discourse upon the word Similitude. And in the eleventh Chapter, wherein you answer to an objection, which might be made to your Argument in the precedent Page, ta­ken from the Parallelism of two concentrique Circles, though the objection be of no moment, yet you have in the same Treatise of yours that which is much more foolish, which is this, Pag. 38. Lin. 12. Non enim magnitu o Anguli, &c. The magnitude of an Angle is not to be estima­ted by that stradling of the legs, which it hath without the Point of concourse, but by that stradling which it hath in the Point of the concourse it self. I pray you tell me what strad­ling there is of two coincident Points, especially such Points as you say are nothing. When did you ever see two nothings straddle?

The Arguments in your twelfth and thirteenth Chapters are grounded all on this untruth, that an Angle is that which is contained between the Lines that make it, that is to say, is a Plain Superficies. Which is manifestly false; because the measure of an Angle is an Arch of a Circle, that is to say, a Line; which is no measure of a Superficies. Besides this gross igno­ [...]ance, your way of Demonstration by putting N for a great Number of sides of an aequilate­rall Polygon, is not to be admitted. For though you understand something by it, you de­monstrate nothing to any Body, but those who understand your Symbolique tongue, which is a very narrow Language. If you had demonstrated it in Irish, or Welsh, though I had not read it, ye [...] I should not have blamed you, because you had written to a considerable Number of mankind, which now you do not.

In your l [...]st Chapters you defend Vitellio without need; for there is no doubt but that whatso­ever crooked Line be touched by a straight Line, the Angle of Contingence will neither add [Page 29] any thing to, nor take any thing from a Rectilineall Right Angle; but that it is because the Angle of Contact is no Angle, or no Quantity, is not true. For it is therefore an Angle, because an Angle of Contact; and therefore Quantity, because one Angle of Contact may be greater then another; and therefore Heterogeneall, because the measure of an Angle of Con­tact cannot (congruere) be applyed to the measure of a Rectilineall Angle, as they think it may, who affirm with you that the Nature of an Angle consisteth in that which is contained between the Lines that comprehend it, viz. in a plain Superficies. And thus you see in how few Lines, and without B [...]achygraphie, your Treatise of the Angle of Contingence is discovered for the greatest part to be false, and for the rest, nothing but a detection of some errors of Clavivs grounded on the same false Principles with your own. To return now from your Treatise of the Angle of Contact back again to your Elenchus.

The fault you find at Art. 18. is, that I understand not that Euclide makes a Plain Angle to be that which is contained between the two Lines that form it. 'Tis true, that I do not un­derstand that Euclide was so absurd, as to think the nature of an Angle to consist in Superficies; but I understand that you have not had the wit to understand Euclide.

The nineteenth Article of mine in this fourteenth Chapter is this; All respect, or variety of Position of two Lines, seemeth to be comprehended in four kinds. For they are either Parallel; or, (being if need be produced) make an Angle; or, (if drawn out faire enough) Touch; or lastly, they are Asymptotes. In which you are first offended with the word It seems. But I allow you that never erre, to be more perempto­ry then I am. For to me it seemed, I say again seemed, that such a Phrase, in case I should leave out something in the enumeration of the severall kinds of Position, would save me from be­ing censured for untruth. And yet your instance of two straight Lines in divers Plains, does not make my enumeration insufficient. For those Lines though not Parallels, nor cutting both the Plaines, yet being moved Parallelly from one Plain to another, will fall into one or other of the kinds of position by me enumerated; and consequently are as much that position, as two straight Lines in the same Plain not parallel, make the same Angle, though not produced till they meet, which they would make if they were so produced. For you have no where proved, nor can prove, that two such Lines do not make an Angle. It is not the actuall concurrence of the Lines, but the Arch of a Circle, drawn upon that point for Center, in which they would meet, if they were produced, and intercepted between them, that constitutes the Angle.

Also your objection con [...]ernin [...] Asymptotes in generall, is absurd. You would have me add, that their distance shall at last be [...] then any distance that can be assigned; and so make the de­finition of the Genus the same with that of the Species. But because you are not Professors of Logick, it is not necessary for me to follow your councell. In like manner, if we under­stand one Line to be moved towards another alwayes parallely to its self, which is, though not actually, yet potentially the same position, all the rest of your instances will come to nothing.

At the two and twentieth Article you object to me the use of the word Figure, before I had defined it: wherein also you do absurdly; for I have no where before made such use of the word Figure, as to argue any thing from it; and therefore your objection is just as wise as if you had round fault with putting the word Figure in the Titles of the Chapters placed before the Book. If you had known the nature of Demonstration, you had not objected this.

You add further, that by my Definition of Figure, a solid Sphere, and a Sphere made hol­low within, is the same Figure; but you say not why, nor can you [...]iver any such thing from my definition. That which deceived your shallowness, is, that you take those Points that are in the concave Superficies of a hollowed Sphere, not to be contiguous to any thing without it, because that whole con [...]ave Superficies is within the whole Sphere. Lastly, for the fault you find, with the definition of like Figures in like positions; I confess there wants the same word which was wanting in the Definition of Paralle [...]s, namely, ad easdem partes (the same way) which should have been added in the end of the definition of like Figures, &c. and may easily be supplyed by any student of Geometry, that is not otherwise a fool.

[Page 30] At the fifteenth Chapter Art. 1. Numb. 6. you object as a contradiction, that I make Mo­tion to be the measure of Time, and yet in other places do usually measure Motion and the affe­ctions thereof by Time. If your thoughts were your own, and not taken rashly out of Books, you could not but (with all men else that see Time measured by Clocks, Dyals, Hour-glasses, and the like) have conceived sufficiently, that there cannot be of Time any other measure be­sides Motion; and that the most universall measure of Motion, is a Line described by some other Motion. Which Line being once exposed to sense, and the motion whereby it was de­scribed sufficiently explicated, will serve to measure all other Motions and their Time; for Time and Motion (Time being but the mentall Image or remembrance of the motion) have but one and the same dimension, which is a Line. But you that would have me measure swiftness and slowness by longer and shorter motion, what do you mean by longer and shorter motion? Is longer and shorter, in the motion, or in the Duration of the motion, which is Time? Or is the Motion, or the Duration of the motion that which is exposed, or design­ed by a Line? Geometricians say often, let the Line A B, be the Time; but never say, let the Line A B be the Motion. There is no unlearned man that understandeth not what is Time, and Motion, and Measure; and yet you that undertake to teach it (most egregious Professors) understand it not.

At the second Article you bring another Argument (which it seems in its proper place, you had forgotten) to prove that a Point is not Quantity not considered, but absolutely Nothing; which is this, That if a Point be not nothing, then the whole is greater then its two halfs. How does that follow? Is it impossible when a Line is divided into two halfs that the middle Point should be divided into two halfs also, being Quantity?

At the seventh Article, I have sufficiently demonstrated, that all Motion is infinitely propagated, as far as space is filled with Body. You alleadge no fault in the demonstration, but object from sense, that the skipping of a Flea, is not propagated to the Indies. If I ask you how you know it, you may wonder perhaps; but answer you cannot. Are you Philoso­phers or Geometricians, or Logicians, more then are the simplest of rurall people? Or are you not rather less, by as much as he that standeth still in ignorance, is nearer to knowledge, then he that runneth from it by erroneous learning?

And lastly, what an absurd objection is it which you make to the eighth Article, where I say that when two Bodies of equall magnitude fall upon a third Body, that which falls with greater velocity, imprints the greater motion? You object, that not so much the magnitude is to be considered as the weight; as if the weight made no difference in the velocity, when notwithstanding weight is nothing else but motion downward? Tell me, when a weighty body thrown upwards worketh on the Body it meeteth with, do you not then think it worketh the more for the greatness, and the less for the weight?

Of the Faults that Occurre in Demon­stration. To the same egregious Professors of the Ma­thematicks in the University of Oxford.

OF twenty Articles which you say (of nineteen which I say) make the six teenth Chap­ter; you except but three, and confidently affirm the rest are false. On the contrary, except three or four faults, such as any Geometrician may see proceed not from igno­norance of the Subject, or from want of the Art of Demonstration, (and such as any man might have mended of himself) but from security; I affirm that they are all true, and truly Demonstrated; and that all your objections proceed from meer ignorance of the Mathe­matiques.

The first fault you find is this, that I express not, (Art. 1.) what Impetus it is, which I would have to be multiplyed into the Time.

The last Article of my thirteenth Chapter was this, If there be a Number of Quantities propounded, howsoever equall or unequall to one another; and there be another Quantity which so often taken as there be Quantities propounded, is equall to their whole sum; that Quantity I call the mean Arithmeticall of them all. Which Definition I did there insert to serve me in the explication of those Propositions of which the sixteenth Chapter consisteth, but did not use it here as I intended. My first Proposition therefore as it standeth yet in the La­tine, being this, The velocity of any Body moved during any Time, is so much as is the product of the Impetus in one Point of Time, multiplyed into the whole Time; to a man that hath not skill enough to supply what is wanting, is not intelligible. Therefore I have caused it in the English to go thus, The velocity of any Body in whatsoever Time moved, hath its Quantity determined by the sum of all the severall (Impetus) Quicknesses, which it hath in the se­verall Points of the Time of the Bodies motion, A [...]d ad [...]ed, that all the Impetus together taken through the whole Time is the same thing with the Mean Impetus (which Mean is de­fined Chapter 13. Art. 29.) multiplyed into the whole Time. To this first Article, as it is un­corrected in the Latine, you object, That meaning by Impetus some middle Impetus, and as­signing none, I determine nothing. And 'tis true. But if you had been Geometricians [...] ­ficient to be Professors, you would have shewed your skill much better, by making it appear that this middle Impetus could be none but that, which being taken so often, as there b [...] Points in the Line of Time, would be equall to the sum of all the severall Impetus taken in the Points of Time respectively; which you could not do.

To the Corollary, you ask first how Impetus can be ordinately applyed to a Line; Absurdly. For does not Ar [...]himedes sometimes say, and with him many other excellent Geometricians, let such a Line be the Time? And do they not mean, that that Line, or the motion over it, is [Page 32] the measure of the Time? And may not also a Line serve to measure the swiftness of a Motion? You thought (you say) onely Lines ought to be said to be ordinately applyed to Lines. Which I easily believe; for I see you understand not that a Line, though it be not the Time it self, may be the quantity of a Time. You thought also all you have, said in your Elenchus, in your Doctrine of the Angle of Contact, in your Arithmetica Infinitorum, and in your Coniques is true; and yet it is almost all proved false, and the rest nothing worth.

Secondly, you object, that I design a Parallelogram by one onely side. It was indeed a great oversight, and argueth somewhat against the man, but nothing against his Art. For he is not worthy to be thought a Geometrician that cannot supply such a fault as that, and correct his Book himself. Though you could not do it, yet another from beyond Sea took notice of the same fault in this manuer, He maketh a Parallelogram of but one side; it should be thus, Ve [...] denique per Parallelogr [...] [...] [...] latus est medium proportionale int [...]r Impetum maxi­mum (five ultimò acquisitum) & impetûs ejusdem maximi semissem; alterum vero latus, medium proportionale, inter totum tempus, & ejus [...]'em totius temporis semissem. Which I therefore repeat, that you may learn good manners; and know, that they who reprehend, ought also, when they can, to add to their reprehension the correction.

At the second Article, you are pleased to advise me, instead of In omni motu uniformi, to put in In omnibus motibus uniformibus. You have a strange opinion of your own Judgement, to think you know to what end another man useth any word, better then himself. My intention was onely to consider motions uniform, and motions from rest uniformly, or regularly accelera­ted, that I might thereby compute the lengths of crooked Lines, such as are described by any of those motions. And therefore it was enough to prove this Theoreme to be true in all uniform or uniformly accelerated Motion, not Motions; though it be true also in the Plurall. It seems you think a man must write all he knows, whether it conduce, or not, to his intended purpose. But that you may know that I was not, (as you think) ignorant how far it might be extended, you may read it Demonstrated at the same Article in the English universally. Against the de­monstration it self you run to another Article, namely, the thirteenth, which is this Probleme, The length being given, which is passed over in a given Time by uniform Motion, to findet e length which shall be passed over by Motion uniformely accelerated in the same Time, so as that the Impetus last acquired be equall to the Time. Which you recite imperfectly, thereby to make it seem that such a Length is not determined. Whether you did this out of ignorance, or on purpose, thinking it a piece of wit, as your pretended mysterie which goes immediately be­fore, I cannot tell, fo [...] in neither place can any wit be espied by any but your selves. To imagine Motions with their Times and Wayes, is a new business, and requires a steddy brain, and a man that can constantly read his own thoughts, without being diverted by the noise of words. The want of this ability, made you stumble and fall unhandsomely in the very first place, (that is in Chap. 13. Art. 13.) where you venture to reckon both Motion and Time at once; and hath made you in this Chapter to stumble in the like manner at every step you go. As for example, when I say, as the product of the Time, and Impetus, to the product of the Time and Impetus, so the Space to the Space when the Motion is Vniform, you come in with nay, rather as the Time to the Time; as if the Parallelograms AI, and AH, were not also as the Times AB, and AF. Thus it is, when men venture upon ways they never had been in before, without a guide.

In the Corollary, you are offended with the permutation of the P [...]ortion of Times and Lines, because you think, you that have scarce one right thought of the Principles of Geometry, that Line, and Time are Heterogeneous Quantities. I know Time and Line are of divers natures; and more, that neither of them is Quantity. Yet they may be both of them Quanta, that is, they may have Quantity; but that their Quantities are Heterogeneous is false. For they are compared and measured both of them by straight lines. And to this there is nothing contrary in the place cited by you out of Clavius; or if there were, 'twere not to be valued. And to your question what is the Proportion of an Hour to an Ell; I answer, it is the same [Page 33] Proportion that two Hours have to two Ells. You see your Question is not so subtile as you thought it By and by you confess that in Times and Lines there is Quid Homogeneum ( [...]s Quid is an infallible sign of not fully understanding what you say) which is false it you [...]ake it of the Lines themselves; though if you take it of their Quantities, it is true without a Quid. Lastly, you tell me how I might have expressed my selfso as it might have been true. But because your ex­pressions please me not, I have not followed your advice.

To the third Article, which is this, In motu uniformiter à quiete accelerato, &c. In mo­tion uniformly acceleratrd from Rest, that is, when the Impetus increaseth in Proportion to the Times, The Length run over in one Time is to the Length run over in another Time, as the Product of the Impetus multiplyed by the Time, to the Product of the Impetus multiplyed by the Time; you object, that the Lengths run over, are in that Proportion which the Impetus hath, to the Impetus; not that which the Impetus hath to the Time, because Impetus to Time has no Proportion, as being Heterogeneous. First, when you say the Impetus, do you mean some one Impetus designed by some one of the unequall straight Lines Parallel to the Base B [...]? That is manifestly false. You mean the aggregate of all those unequall Parallels. But that is the same thing with the Time multiplyed into the mean Impetus. And so you say the same that I do. Agai [...], I ask where it is that I say or dream that the Lengths run over are in the Proporti­on of the Impetus to the Times? Is it you or I that dream? And for your Heterogeneity of the Quantities of Time and of Swiftness, I have already in divers places shewed you your error. Again, Why do you make BI represent the Lengths run over, which I make to be represented by DE, a Line taken at pleasure; and you also a few Lines before make the same BI to design the greatest acquired Impetus? These are things which shew that you are puzzled and intangled with the unusuall speculation of Time and Motion, and yet are thrust on with Pride and Spi [...]e to adventure upon the examination of this Chapter.

Secondly, you grant the Demonstration to be good, supposing I meane it (as I seeme to speak) of one and the same Motion. But why doe I not meane it of one and the same Motion, when I say not in Motions, but in Motion uniform? Because (say you) in that which follows, I draw it also to different Motions. You should have given at least one Instance of it; but there is no such matter. And yet the Proposition, is in that case also true; though then it must not be Demonstrated by the similitude of Triangles, as in the case present. And therefore the ob­jections you make from different Impetus acquired in the same time, and from other cases which you mention, are nothing worth.

At the fourth Article, you allow the Demonstration all the way (except the faults of the third, which I have already proved to be none) till I come to say, that because the Proportion of FK to BI is double to the Proportion of AF to AB, therefore the Proportion of AB to AF is double to the Proportion of BI to FK. This you deny, and wonder at, as strange (for it is indeed strange to you) and in many places you exclaim against it as extream Igno­rance in Geometry. In this place you onely say, No such matter; for though one Proportion be double to another, yet it does not follow that th [...] Converse is the double of the Converse. So that this is the issue to which the Question is reduced, whether you have any or no Geometry. I say, if there he three Quantities in continuall Proportion, and the first be the least, the Pro­portion of the first to the second is double to the Proportion of the first to the third; and you deny it. The reason of our dissent consisteth in this, that you think the doubling o [...] a Propor­tion to be the doubling of the Quantity of the Proportion, as well in Proportions of Defect, [...]s in Proportions of Excess; and I think that the doubling of a Proportion of Defect, is the doubling of the defect of the Quantity of the same. As for example in these three numb [...]s, 1, 2, 4, which are in continuall Proportion, I say the Quantity of the Proportion o [...] one to two, is double the Quantity of the Proportion of one to four. And the Quantity of the Proportion of one to four, is half the Quantity of the Proportion of one to two. And yet deny not but that the Quantity of the Defect in the Proportion of one to two is doubled in the Proportion of one to four. But because the doubling of defect makes greate [...] defect, it [Page 34] maketh the Quantity of the Proportion less. And as for the part which I hold in this Que­stion, first, there is thus much demonstrated by Euclide, E [...]. 5. Prop. 8; that the Propor­tion of one to two is greater then the Proportion of one to four, though how much it is greater be not there Demonstrated. Secondly, I have Demonstrated (Chap. 13. A [...]t. 16.) That it is twice as great, that is to say, (to a man that speaks English) double. The introducing of duplicate, triplicate, &c. instead of double, triple, &c. (thoug [...] now they be words well un­derstood by such as understand what Proportion is) proceeded at first from such as durst not for fear of absurdity, call the half of any thing double to the whole, though it be manifest that the half of any defect is a double Quantity to the whole defect; [...]or want added to want maketh greater want, that is, a less positive Quantity. This difference between double, and duplicate, lighting upon weak understandings, h [...]s put men out of the way of true reasoning in very many Questions of Geometry. Euclide never used but one word both for double and duplicate. It is the same fault when men call half a Quantity Subduplicate, and a third pa [...]t Subtriplicate of the whole, with intention (as in this case) to make them pass for words of signification diffe­rent from the half and the third part. Besides, from my Definition of Proportion (which is clear, and easie to be understood by all men, but such as have read the Geometry of others un­luckily) I can Demonstrate the same evidently and briefly thus. My Definition is this, Pro­portion is the Quantity of one Mag [...]itude taken comparatively to another. Let there be there­fore three Quantities, 1, 2, 4, in continuall Proportion. Seeing therefore the Quanti­tity of four in respect of one, is twice as great as the Quantity of the same four in respect of 2, it followeth manifestly that the Quantity of 1 in respect of 4, is twice as little as the Quantity of the same 1 in respect of 2; and consequently the Quantity of one in respect of two, is twice as great as the Quantity of the same one in respect of four; which is the thing I maintain in this Question Would not a man that imployes his time at Bowles, choose rather to have the advantage given him of three in nine, then of one in nine? And why, but that three is a greater Quantity in respect of nine, then is one? Which is as much as to say, three to nine hath a greater Proportion then one to nine; as is Demonstrated by Euclide, El. 5. [...]. Is it not therefore (you that profess Mathematiques, and Theology, and practise the depression of the truth in both) well ow [...]'d of you, to teach the contrary? But where you say that the Point K (in the second Figure of the Table belonging to this 16. Chapter) is not in the Parabolicall Line whose Diameter is AB, and Base BI, but in the Parabolicall Line of the Complement of my Semiparabola (as I may learn from the twenty-third Proposition of your Arithmetica Infinitorum) whose Diameter is AC, and Base IC. What Line is that? Is it the same Line with that of my Semiparabola, or not the same? If the same, why find you fault? If not the same, you ought to have made a Semiparabola on the Diameter AC, and Base IC, and following my Const [...]uction made it appear that K is not in the Line wherein I say it is; which you have not done, nor could do.

Then again, running on in the same blindness of Passion, you pretend I make the Propor­tion of BI to FK double to that of AB to AF, and then confute it; when you knew I made the Proportion of FK to BI, double to that of FN, to BI, that is, of AF to AB; and this was it you should have confuted. That which followeth is but a Triumphing in your own Ig­norance, wherein you also say, That all that I afterwards build upon this Doctrine is false. You see whether it be like to prove so or not. As for your Arithmetica Infinitorum, I shall then read to you a piece of a Lesson on it when I come to your objections against the next Chapter. In the mean Time let me tell you, it is not likely you should be great Geometricians, that know not what is Quantity, nor Measure, nor Straight, nor Angle, nor Homogeneous, nor Heterogeneous, nor Proportion, as I have already made appear in this and the former Lessons.

To the first Corollary of this fourth Article your exception I confess is just, and (which I wonder at) without any incivility. But this argues not Ignorance, but Security. For who is there that ever read any thing in the Coniques, that knows not that the parts of a Parabola [Page 35] cut off by Lines Parallel to the Base, are in Triplicate Proportion to their Bases? But having hitherto designed the Time by the Diameter, and the Impetus by the Bas [...]; and in the next Chapter (where I was to calculate the Proportion of the Parabola, to the Parallelogram) inten­ding to design the Time by the Base, I mistook and put the Diameter again for the Time; which any man but you might as easily have corrected as reprehended.

To the second Corollarie, which is this, That the Lengths run over in equall Times by Motion so accelerated, as that the Imperus increase in double Proportion to their Times, are as the differences of the Cubick numbers beginning at unity, that is, as seven, nineteen, thirty-se­ven, &c. You say 'tis false. But why? Because (say you) portions of the Parabola of equall altitude taken from the beginning are not as those numbers seven, nineteen, thirty-seven, &c. Does this think you, contradict any thing in this Proposition of mine? Yes, because (you think) the lengths gone over in equall Times, are the same with the parts of the Diameter cut off from the Vertex, and proportionall to the numbers one, two, three, &c. Whereas the lengths run over, are as the aggregates of their velocities, that is, as the parts of the Parabola itself, that is, as the Cubes of their Bases, that is, as the numbers one, eight, twenty-seven, sixty-four, &c. and consequently the lengths run over in equall Times, are as the differences of those Cubick Numbers one, eight, twenty-seven, sixty-four, whose differences are seven, nineteen, thirty-se­ven, &c. The cause of your mistake was, that you cannot yet, nor perhaps ever will contempla [...]e Time and Motion (which requi [...]eth a steddy b [...]ain) without confusion.

The third Corollary, you also say is false, Whether it be meant of Motion uniformly accele­rated (as the words are) or (as perhaps, you say, I meant it) of such Motion as is accelerated in double Proportion to the Tim [...]. You need not say perhaps I meant it. The words of the Proposition are enough to make t [...]e meaning of the Corollary understood. But so also you say it is false. Me thinks you should have offered some little proof to make it seem so. You think your Authority will carry it. But on the contrary I believe rather that they that shall see how your other objections hitherto have sped, will the rather think it true, because you think it false. The Demonstration as it is, is evident enough; and therefore I saw no cause to change a word of it.

To the fifth Article you object nothing, but that it dependeth on this Proposition (Chap. 13. Art. 16.) T [...]at when thr [...]e Quantities are in continuall Proportion, and the first is the least, as in these numbers, four, six, nine. The Proportion of the first to the second is dou­ble to the Proportion of the same first to the last. Which is there demonstrated, and in the former Lessons so amply explicated, as no man can make any further doubt of the truth of it. And you will, I doubt not as [...]ent unto it. But in what estate of mind will you be then? A man of a tender forehead after so much insolence, and so much contumelious language grounded upon arrogance and ignorance, would hardly indure to out-live it. In this vanity o [...] yours, you ask me whether I be angry, or blush, or can endure to hear you. I have some reason to be an­gry; for what man can be so patient as not to be moved with so many injuries? And I have some reason to blush, considering the opinion men will have beyond Sea, (when they shall see this in Latine) of the Geometry taught in Oxford. But to read the worst you can say against me, I can indure, as easily at least, as to read any thing you have written in your Treatises of the Angle of Co [...]tact, of the Conique-sections, or your Arithmetica Infinitorum.

The sixth, seventh, eighth Articles, you say are sound. True. But never the more to be thought so for your app [...]obation, but the less; because you are not fit, neither to reprehend, nor praise; and because all that you have hither to condemned as fals [...], hath been proved true. Then you shew me how you could demonstrate the sixth and seventh Articles a shorter way. But though there be your Symboles, yet no man is obliged to take them for demonstration. And though they be granted to be dumb Demoostrations, yet when they are taught to speak as they [...]ught to do, they will be longer Demonstrations then these of mine.

To the nineth Article, which is this, If a Body be moved by two Movents at once, concur­ring in what Angle so [...]ver, of which, one is moved uniformly, the other, with Motion uni­formly [Page 36] accelerated from Rest, till it acquire an Impetus equall to that of the uniform motion, the line in which the Body is carried, shall be the crooked line of a Semiparabola. You lift up your voice again, and ask what Latitude? what Diameter? what Inclination of the Diame­ter to the Ordinate Lines? If your Founder should see this, or the like objections of yours, he would think his money ill bestowed. When I say in what Angle soever, you ask, in what Angle? When I say two Movents, one unif [...]rm, the other uniformly accelerated, make the Body describe a Semiparabolicall line; you ask, which is the Diameter; as not knowing that the accelerated motion describes the Diameter, and the other a Parallel to the Base. And when I say the two Movents meet in a Point, form which Point both the Motions begin, and one of them from Rest, you ask me, what is the Altitude? As if that Point where the Motion be­gins from Rest were not the Vertex; or that the Vertex and Base being given, you had not wit enough to see that the Altitude of the Parabola is determined? When Galileo's Proposition, which is he same with this of mine, supposed no more but a Body moved by these two Motions, to prove the Line described to be the crooked Line of a Semiparabola, I never thought of ask­ing him what Altitude, nor what Diameter, nor what Angle, nor what Base had his Parabola. And when Archimedes said, let the Line A B be the Time, I should never have said to him, do you think Time to be a Line, as you ask me whether I think Impetus can be the Base of a Pa­rabola. And why, but because I am not so egregious a Mathematician, as you are. In this giddiness of yours, caused by looking upon this intricate business of Motion, and of Time, and the concourse of Motion uniform, and uniformly accelerated, you rave upon the numbers 1, 4, 9, 16, &c. without reference to any thing that I had said; insomuch as any one that had seen how much you have been deceived in them before in your scurvy Book of Arithmetica Infinitorum, would presently conclude, that this objection was nothing else but a fit of the same madness which possest you there.

My tenth Article is like my nineth; and your objections to it are the same which are to the former. Therefore you must take for answer just the same which I have given to your objection there.

To the eleventh, you say first, you have done it better at the sixty-fourth Article of your Arithmetica Infinitorum. But what you have done there, shall be examined when I come to the Defence of my next Chapter. And whereas I direct the Reader for the finding of the Pro­portions of the Complements of those Figures to the Figures themselves, to the Table of Art. 3. Chap. 17. you say that if the encrease of the Spaces, were to the encrease of the Times, as one to two, then the complement should be to the Parallelogram as one to three, and say you find not ⅓ in the Table. Did you not see that the Table is onely of those Figures which are descri­bed by the concourse of a Motion uniform with a motion accelerated? You had no reason there­fore to look for ⅓ in that Table; for your case is of Motion uniform concurring with Motion retarded, because you make not the Proportions of the Spaces to the Proportions of the Times as two to one, but the contrary; so that your objection ariseth from want of observing what you read. But I may learn (you say) these, and greater Matters then these, in your twenty-third and sixty-fourth Propositions of your Arithmetica Infinitorum. This which you say here is a great Absurdity; but if you mean I shall finde greater there, I will not say against you. This ⅓ you looked for, belongs to the Complements of the Figures calculated in that Table; which because you are not able to find out of your selves, I will direct you to them. Your case is of ⅓ for the Complement of a Parabola. Take the Denominator of the Fraction which belongs to the Parabola, namely three, and for Numerator take the Numerator of the Fraction which be­longs to the Triangle, namely one, and you have the Fraction sought. And in like manner for the [Page 37] Complement of any other Figuer. As for example, of the second Parabolaster, whose Fracti­on hath for Denominator five, take the Numerator of the Fraction of the same Triangle which is one, and you have ⅕ for the Fraction sought for; and so of the rest, taking alwayes one for the Numerator.

The twelfth Article, which you say is miserably false, I have left standing unaltered. For not comprehending the sense of the Proposition, you make a Figure of your own, and fight against your own fancied Motions, different from mine. Other Geometricians that understand the construction better, find no fault. And if you had in your own fifth Figure drawn a Line through N Parallel to A E, and upon that Line supposed your accelerated Motion, you would quickly have seen that in the Time AE, the Body moved from Rest in A, would have fallen short of the Diagonall A D; and that all your extravagant pursuing of your own mistake had been absurd.

My thirteenth Article you say is ridiculous. But why? The Impetus last acquired cannot (you say) be equall to a Time. But the Quantity of the Impetus may be equall to the Quan­tity of a Time, seeing they are both measured by Line. And when they are measured by the same described Line, each of their Quantities is equall to that same Line, and consequently to one another. But when I meet with this kind of objection again, since I have so often already shewn it to be frivolous, and no l [...]ss to be objected against all the Antients that ever Demon­strated any thing by Motion, then against me, I purpose to neglect it.

Secondly, you object That Motion uniformly accelerated does no more determine Swiftness, then Motion uniform. True; you needed not have used sixteen Lines to set down that. But suppose I add (as I do) so as the last acquired Impetus be equall to the Time. But that (you say) is not sense; Which is the objection I am to neglect. But (you say again) supposing it sense, this limitation helps me nothing. Why? Because (you say) a Parabola may be descri­bed upon a Base given, and yet have any Altitude, or any Diameter one will. Who doubts it? But how follows it from thence, that when a Parabolicall Line is described by two Motions, one uniform, the other uniformly accelerated from Rest, that the determining of the Base does not also determine the whole Parabola? But fifthly (you say) that this equality of the Impetus to the Time does not determine the Base. Why not? Because (you say) it is an error proceed­ing from this, that I understand not what is Ratio subduplicate. I look't for this. I have shewn and inculcated sufficiently before, that the error is on your side; and therefore must tell you, that this objection, and also a great part of the rest of your errors in Geometry proceedeth from this, that you know not what Proportion is. But see how wisely you argue about this du­plication of Proportion. For thus you say verbatim. Stay a little. What Proportion has duplicate Proportion to single Proportion? Is it alwayes the same? I think not. For ex­ample. Duplicate Proportion [...] is double to the single 2/1. Duplicate Proportion [...] is triple to its single 3/1. Let any man, even of them that are most ready in your Symboles, say in your behalf (if he be not ashamed) that the Proportion of nine to one is triple to the Proportion of three to one, as you do.

In the fourteenth, fifteenth, and sixteenth Article, you bid me repeat your objections to the thirteenth. I have done it; and find that what you have objected to the thirteenth, may as well be objected to these; and consequently, that my answer there will also serve me here. There­fore (if you can endure it) read the same answer over again.

But you have not yet done (you say) with these Articles. Therefore (after you had for a while spoken perplextly, conjecturing not without just cause, that I could not understand you) you say that to the end I may the better perceive your meaning, I should take the example follow­ing. Let a Movent (in the first Figure of this Chapter) be moved uniformly, in the Time A B, with the continuall Impetus A C, or B I, whose whole velocity shall therefore be the [Page 38] Parallelogram A C I B. And another Movent be uniformly accelerated, so as in the Time A B it acquire the same Impetus B I. Now as the whole velocity, is to the whole velocity, so is the length run over, to the length run over. All this I acknowledge to be according to my sense, saving that your putting your word Movens instead of my word Mobile hath corrupted this Article. For in the first Article, I meddle not with Motion by Concourse, wherein only I have to do with two Movents to make one Motion; but in this I do, wherein my word is not Movens but Mobile; by which it is easie to perceive you understand not this Proposition. Then you proceed, But the length run over by that accelerated Motion is greater then the length run over by that unifo [...] Motion. Where do I say that? You answer, in the nineth and thir­teenth Article, in making [...] (in the fifth Figure) greater then A C; and A H (in the eighth Figure) greater then A B; and consequently, the Triangle A B I, greater then the Parallelogram A C I B. That consequently is without consequence; for it importeth no­thing at all in this Demonstration, whether A B, or A C in the fifth Figure be the greater. Besides, I speak there of the Concourse of two Movents that describe the Parabolicall Line A G D; where the increasing Impetus, (because it increaseth as the Times) will be designed by the Ordinate Lines in the Parabola A G D B. And if both the Motions in A B and A C were uniform, the Aggregate of the Impetus would be designed by the Triangle A B D, which is less then the Parallelogram A C D B. But you thought that the Motion made by A C uniformly, is the same with the Motion made uniformly in the same Time by the Motions in A B, and A C, concurring. So likewise in the eighth Figure, there is nothing hinders A H from being greater then A B, unless I had said that A B had been described in the Time A C with the whole Impetus A C maintained entire; of which there is nothing in the Propo­sition, nor would at all have been pertinent to it. Therefore all this new undertaking of the thirteenth, fourteenth, fifteenth and sixteenth Articles, is to as little purpose as your former objections. But I perceive that these new and hard Speculations, though they turn the edge of your wit, turn not the edge of your malice.

At the seventeenth Article, you shew again the same confusion. Return to the eighth Fi­gure, If in a Time given a Body run over two lengths, one with uniform, the other with ac­celerated Motion; as for example, if in the same Time A C a Body run over the Line A B with uniform Motion, and the Line A H with Motion accelerated; and again in a part of that Time it run over a part of the length A H, with uniform Motion, and another part of the same with Motion accelerated; as for example, in the Time A M it run over with uniform Motion the Line A I, and with Motion accelerated the Line A B. I say the excess of the whole A H above the part A B, is to the excess of the whole A B above the part A I, as the whole A H, to the whole A B. But first you will say, that these words as the whole A H to the whole A B, are left out in the Proposition. But you acknowledge that it was my meaning; and you see it is expressed before I come to the Demonstration. And therefore it was absurdly done to reprehend it. Let us therefore pass to the Demonstration. Draw I K Parallel to A C, and make up the Parallelogram A I K M. And supposing first the acceleration to be uniform, divide I K in the midst at N; and between I N, and I K, take a mean Proportionall I L. And the straight Line A L, drawn and produced, shall cut the Line B D in F, and the Line C G in G (which Lines C G, and B D, as also H G and B F, are determined (though you could not carry it so long in memory) by the Demonstration of the thirteen Article.) For seeing A B is described by Motion uniformly accelerated, and A I by Motion uniform to the same Time A M; and I L is a mean Proportionall between I N (the half of I K) and I K; therefore by the Demonstration of the thirteenth Article; A I is a mean Proportionall between A B and the half of A B, namly A O. Again, because A B is described by uniform Motion, and A H by Motion uniformly accelerated, both of them in the same Time A C, B F is a mean Proportionall betwen B D and half B D, namely B E; Therefore by the Demonstration of the same thirteenth Article, the straight Line A L F produced will fall on G; and the Line A H will be to the Line A B, as the Line A B to the Line A I. And consequently as A H to A B, [Page 39] so H B to B I; which was to be Demonstrated. And by the like Demonstration the same may be proved, where the acceleration is in any other Proportion that can be assigned in Num­bers, saving that whereas this Demonstration dependeth on the construction of the thirteenth Article, it the Motion had been accelerated in double Proportion to the Times, it would have depended on the fourteenth, where the Lines are determined. Which determinations being not repeated, but declared before in the thirteenth Article, to which this Diagram belongeth, you take no notic [...] of, but go back to a Figure belonging to another Article where there was no use of these Determinations. But because I see that the words of the Proposition, are as of four Motions, and not of two Motions made by twice two Movents, I must pardon them that have not righ [...]ly understood my meaning; and I have now made the Proposition according to the Demonstration. Which being done, all that you have said in very neer two leaves of your Elenchus comes to nothing; and the fault you find comes to no more then a too much trusting to the skill and diligence of the Reader. And whereas after you had sufficiently trou­bled your self upon this occasion, you add, that if Sir H. Savile had read my Geometry, he had never given that censure of Joseph Scaliger, in his Lectures upon Euclide, that he was the the worst Geometrician of all Mortall men, not excepting so much as Orontius, but that praise should have been kept for me; You see by this time, at least others do, how little I ought to value that opinion; and that though I be the least of Geometricians, yet my Geo­metry is to yours as 1 to 0. I recite these words of yours, to let the world see your indiscretion in mentioning so needlesly that passage of your Founder. It is well known that Joseph Scaliger deserved as well of the state of Learning, as any man before or since him; and that though he failed in his Ratiocination concerning the Quadrature of the Circle, yet there appears in that very failing so much knowledge of Geometry, that Sir H. Savile could not but see that there were mortall men very many that had less; and consequently he knew that that cen­sure of his in a rigid sense (without the License of an Hyperbole) was unjust. But who is there that will approve of such Hyperboles to the dishonour of any but of unworthy persons, or think Joseph Scaliger unworthy of honour from Learned men? Besides, it was not Sir H. Savile that confuted that false Quadrature, but Clavius. What honour was it then for him to triumph in the victory of another? When a beast is slain by a Lion, is it not easie for any of the Fowles of the Air to settle upon, and peck him? Lastly, though it were a great error in Scaliger, yet it was not so great a fault as the least Sin; and I believe that a publique contumely done to any worthy person after his death, is not the least of Sins. Judge therefore whether you have not done indiscreetly, in reviving the onely fault, perhaps that any man li­ving can lay to your Founders charge; and yet this error of Scaligers was no greater then one of your own of the like nature, in making the true Spirals of Archimedes equall to half the circumference of the Circle of the first revolution; and then thinking to cover your fault by calling it afterwards an Aggregate of Arches of Circles (which is no Spirall at all of any kind) you do not repair but double the absurdity. What would Sir Henry Savile have said to this?

The eighteenth Article is this, In any Parallelogram, if the two sides that contain the An­gle be moved to their opposite sides, the one uniformly, the other uniformly accelerated; the side that is moved uniformly, by its concourse through all its Longitude, hath the same effect which it would have if the other Motion were also uniform, and the Line described, were a mean Proportionall between the whole Length, and the half of the same.

To the Proposition you object first, that it is all one whether the other motion be uniform or not, because the effect of each of their Motions, is but to carry the Body to the opposite side. But do you think that whatsoever be the Motions, the Body shall be carryed by their concourse alwayes to the same Point of the opposite side? If not, then the effect is not all one when a Motion is made by the concourse of two Motions uniform and accelerated, and when it is made by the concourse of two uniform or of two accelerated Motions.

Secondly, you say that these words, and the Line described were a mean Proportionall be­tween [Page 40] the whole Length, and the half of the same, have no sense, or that you are deceived. True. For you are deceived; or rather you have not understanding enough distinctly to con­ceive variety of Motions though distinctly expressed. For when a Line is gone over with Motion uniformly accelerated, you cannot understand how a mean Proportionall can be taken between it and its half; or if you can, you cannot conceive that that mean can be gone over with uni­form motion in the same Time that the whole Line was run over by motion uniformly acce­lerated. Yet these are things conceivable, and your want of understanding must be made my fault.

My Demonstration is this, In the Parallelogram A B C D, (Fig. 11.) Let the side A B be conceived to be moved uniformly till it lye in C D; and let the Time of that Motion be A C, or B D. And in the same Time let it be conceived that A C is moved with uniform accelerati­on, till it lye in B D. To which you object, that then the acceleration last acquired must be far greater then that wherewith A B is moved uniformly; else it shall never come to the place you would have it in the same Time. What proof bring you for this? None here. Where then? No where that I remember. On the contrary I have proved (Art. 9. of this Chap­ter) that the Line described by the concourse of those two Motions, namely, uniform from A B to C D, and uniformly accelerated from A C to B D, is the crooked Line of the S [...] ­miparabola A H D. And though I had not, yet it is well known that the same is demonstra­ted by Galileo. And seeing it is manifest that in what Proportion the Motion is accelerated in the Line A B, in the same Proportion the Impetus beginning from Rest in A is encreased in the same Times (which Impetus is designed all the way by the ordinate Lines of the Semi-Parabola) the greatest Impetus acquired must needs be the Base of the Semiparabola, namely B D, equal to A C which designs the whole Time. I cannot therefore imagine what should make you say without proof, that the greatest acquired Impetus is greater then that which is designed by the Base B D. Next you say, you see not to what end I divide A B in the middle at E. No wonder; for you have seen nothing all the way. Others would s [...] it is necessary for the De­monstration; as also that the Point F is not to be taken arbitrarily; and likewise that the thirteenth Article (which you admit not for proof) is sufficiently demonstrated, and your ob­jections to it answered. By the way you advise me, where I say percursam codem motu unifor­mi, cum Impetu ubi (que) &c. to blot out cum; because the Impetus is not a companion in the way, but the cause. Pardon me in that I cannot take your Learned Counsell; for the word motu uniformi is the Ablative of the Cause, and Impetu the Ablative of the Manner. But to come again to your objections, you say I make a greater space run over in the same Time by the slower Motion then by the swifter. How does that appear? because there is no doubt, but the swiftness is greater where the greatest Impetus is alwaies maintained, then where it is attained to in the same Time from Rest. True, But that is when they are considered asunder without concourse, but not then when by the concourse they debilitate one another, and describe a third Line different from both the Lines, which they would describe singly. In this place I compare their effects as contributing to the the description of the Parabolicall Line A H D. What the effects of their severall Motions are, when they are considered asunder, is sufficiently shewn b [...]fore in the first Article. You should first have gotten into your mindes the perfect and distinct Ideás of all the Motions mentioned in this Chapter, and then have ventured upon the censure of them, but not before. And then you would have seen that the Body moved from A, describeth not the Line A C, nor the Line A B, but a third, namely the Semiparabolicall Line A H D.

Again, where I say, Wherefore if the whole A B be uniformly moved to C D, in the same Time wherein A C is moved uniformly to F G; you ask me whether with the same Impe­tus or not. How is it possible that in the same time two unequall Lengths should be passed over with the same Impetus? But why (say you) do you not tell us with what Impetus A C comes to F G? What need is there of that, when all men know that in unifo [...]m Motion and the same Time, Impetus is to Impetus, as Length to Length? Which to have expressed had not [...]een pertinent to the Demonstration. That which follows in the Demonstration rursus sup­pono [Page 41] quod latus A C, &c. to these words, ut ostensum est, Art. 12, You confute with saying you have proved that Article to be false. But you may see now, if you please, at the same place that I have proved your objections to be frivolous.

After this you run on without any Argument against the rest of the Demonstration, shew­ing nothing all the way, but that the variety and concourse of Motions, the Speculations whereof you have not been used to, have made you giddy.

To the nineteenth Article you apply the same objection which you made to the eighteenth. Which having been answered, it appeares that from the very beginning of your Elenchus to this place all your objections (except such as are made to three or four mistakes of small importance in setting down my mind,) are meer Paralogisms, and such as are less pardonable then any Pa­ralogism in Orontius, both because the Subject as less difficult is more easily mastered, and be­cause the same faults are more shamefully committed by a Reprehender then by any other man.

I had once added to these nineteen Articles a twentieth, which was this, If from a Point in the Circumference there be drawn a Chord, and a Tangent equall to it, the Angle which they make shall be double to the aggregate of all the Angles made by the chords of all the equal Arches into which the Arch given canpossibly be divided. Which Proposition is true, and I did when I writ it think I might have use of it. But be it, or the Demonstration of it true or false, seeing it was not published by me, it is somewhat barbarous to charge me with the faults thereof. No Doctor of Humanity but would have thought it a poor and wretched malice, publiquely to examine and cen­sure papers of Geometry never published, by what means soever they came into his hands. I must confess that in these words, in such kind of Progression Arithmeticall (that is, which begins with o) the sum of all the Numbers taken together, is equall to half the Number that is made by multiplying the greatest into the least, there is a great error; for by this account these Num­bers, 0, 1, 2, 3, 4, taken together should be equall to nothing. I should have said they are equall to that Number which is made by multiplying half the greatest into the Number of the Terms. There was therefore, if those words were mine (for truly I have no Copy of them, nor have had since the Book was Printed, and I have no great reason, as any man may see, to trust your Faith) a great error in the writing, but not an erroneous opinion in the writer. The Demonstration so corrected is true. And the Angles that have the Propor­tions of the Numbers 1, 2, 3, 4, are in the Table of your Elenchus, Fig. 12. the Angles G A D, H D E, I E F, K F B. And if the Divisions were infinite, so that the first were not to be reckoned but as a cypher, the Angle C A B would be double to them all together. This mistake of mine, and the finding that I had made no use of it in the whole book, was the cause why I thought fit to leave it quite out. But your Professorships, could not forbear to take occasion thereby, to commend your zeal against Leviathan to your Doctorships of Divi­nity, by censuring it.

Of the Faults that Occurre in Demon­stration. To the same egregious Professors of the Ma­thematicks in the University of Oxford.

AT the seventeenth Chapter, your first exception is to the Definition of Proportionall Prop [...]rtions, which is this, Four Proportions are then Proportionall, when the first is to the second, as the third to the fourth. The Reader will hardly believe that your ex­ception is in earnest. You say I mean not by Proportionality the Quantity of the Proporti­ons. Yes I do, Therefore I say again, that four Proportions are then Proportionall, when the Quantity of the first Proportion, is to the Quantity of the second Proportion, as the Quantity of the third Proportion, to the Quantity of the fourth Proportion. Is not my meaning now plainly enough expressed? Or is it not the same Definition with the former? But what do I mean (you will say) by the Quantity of a Proportion? I mean the determined great­ness of it, that is, for example, in these Numbers, the Quantity of the Proportion of two to three, is the same with the Quantity of the Proportion of four to six, or six to nine; and a­gain, the Quantity of the Proportion of six to four, is the same with the Quantity of the Pro­portion of nine to six, or of three to two. But now what do you mean by the Quantity of a Proportion? You mean that two and three, are the Quantities of the Proportion of two to three (for so Euclide calls them) and that six and four are the Quantities of the Proportion of six to four, which is the same with the Proportion of three to two. And by this Rule, one and the same Proportion shall have an infinite Number of Quantities; and consequently the Quantity of a Proportion can never be determined. I call one Proportion double to another, when one is equall to twice the other; as the Proportion of four to one, is double to the Pro­portion of two to one. You call that Proportion double where one Number, Line, or Quan­tity absolute is double to the other; so that with you the Proportion of two to one is a double Proportion. It is easie to understand how the Number two is double to one, but to what I pray you, is double the Proportion of two to one, or of one to two? Is not every double Pro­portion double to some Proportion? See whether this Geometry of yours can be taken by any man of sound mind for sense. But 'tis known (you say) that in Proportions double is one thing, and duplicate another; So that it seems to you, that in talking of Proportion men are allowed to speak senslessly. 'Tis known, you say. To whom? It is indeed in use at this day to call jouble duplicate, and triple triplicate. And it is well enough; for they are words that signifie the same thing, But that they differ (in what subject soever) I never heard till now. I am sure that Euclide whom you have undertaken to expound, maketh no such difference. And even there where he putteth these Numbers, one, two, four, eight, &c. for Numbers in double Proportion (which is the last Proposition of the nineth Element) he meaneth not that one to [Page 43] two, or two to one, is a double Proportion, but that every Number in that Progression is dou­ble to the number next before it; and yet lie does not call it Analogia dupla, but Duplicata. This distinction in Proportions between double and duplicate, proceeded long after from want of knowledge that the Proportion of one to two is double to the Proportion of one to four; and this from ignorance of the different nature of Proportions of Excess, and Proportions of De­fect. And you that have nothing but by tradition saw not the absurdities that did hang thereon.

In the second Article I make E K, (Fig. 1.) the third part of L K, which you say is false; and consequently the Proposition undemonstrated. And thus you prove it false. Let A C be to G C, or G K to G L, as eight to one (for seeing the Point G is taken arbitrarily, we may place it where w [...] will, &c.) and upon this placing of G arbitrarily, you prove well enough that E K is not a third part [...]f L K. But you did not then observe, that I make the Altitude A G, less then any Qu [...]ntity given, and by consequence E K to differ from a third part by a less dif­ference then any Quantity that can be given. Therefore as yet the Demonstration proccedeth well enough. But perceiving your oversight, you thought fit (though before, you thought this confutation sufficient) to endeavour to confure it another way; but with much more evidence of ignorance. For when I come to say, the Proportion therefore between A C and G C is triple (in Arithmeticall Proportion) to the Proportion between G K and G E, &c. you say the Proportion of A C to G C, is the Proportion of Identity, as also that of G K to G E. But why? Does my construction make it so? Do not I make G C less then A C, though with less Difference then any Quantity that can be assigned? And then where I say therefore E K is the third part of L K, you come in (by Parenthesis) with (or a fourth, or a fifth &c.) Upon what ground? Because you think it will pass for current, without proof, that a Point is nothing. Which if it do, Geometry also shall pass for nothing, as having no ground nor begin­ning but in nothing. But I have already in a former Lesson sufficiently shew'd you the conse­quence of that opinion. To which I may add, that it destroys the method of Indivisibles, invented by Bonaventura; and upon which, not well understood, you have grounded all your scurvy book of Arithmetica Infinitorum; where your Indivisibles have nothing to do, but as they are supposed to have Quantity, that is to say, to be Divisibles. You allow, it seems, your own nothings to be somethings, and yet will not allow my somethings to be considered as no­thing. The rest of your objections having no other ground then this, that a Point is nothing, my whole Demonstration standeth firm; and so do the Demonstrations of all such Geometricians, Ancient and Modern, as have inferred any thing in the manner following, viz. If it be not greater nor less, then it is equall. But it is neither greater nor less. Therefore, &c. If it be greater, say by how much. By so much. 'Tis not greater by so much. There­fore it is not greater. If it be less, say by how much, &c. Which being good Demonstrati­ons are together with mine overthrown by the nothingness of your Point, or rather of your un­standing; upon which you nevertheless have the vanity of advising me what to do, if I demon­strate the same again; meaning I should come to your false, impossible, and absurd Method of Arithmetica Infinitorum, worthy to be gilded, I do not mean with Gold.

And for your Question, why I set the Base of my Figure upwards, you may be sure it was not because I was affraid to say, that the Proportions of the ordinate Lines beginning at the Ver­tex were triplicate, or otherwise multiplicate of the Proportions of the intercepted parts of the Diameter. For I never doubted to call double duplicate, nor triple triplicate, &c. or if I had, I should have avoided it afterwards at the tenth Article of the same Chapter. But because when I went about t [...] compare the Proportions of the ordinate Lines with those of their contiguous Diameters, the first thing I considered in them was in what manner the Base grew less and less till it vanished into a Point. And th [...]ugh the base had been placed below, it had not therefore re­quired any change in the Demonstration. But I was the more apt to place the Base uppermost, because the Motion began at the Base, and ended at the Vertex. To proceed which way I pleased was in my own choice; and it is of grace that I give you any account of it at all.

[Page 44] To the third Article together with its Table, you say, it falls in the ruine of the second; and that the same is to be understood of the sixth, seventh, eighth and nineth. For confutation whereof I need to say no more, but that they all stand good by the confutation of your objecti­ons to the second.

To the fourth Article you say, the description of those curvilineall Figures is easie. True, to some men; and now that I have shewed you the way, 'tis easie enough for you also. For the way you propound is wholly transcribed out of the Figure of the second Article, which Ar­ticle you had before rejected. For seeing the Lines HF, GE, AB, &c. are equall to the Lines CQ, CO, CD; and the Lines QF, OE, DB, equall to the Lines CH, CG, CA; the Proportion of DB to OE, will be triple (that is, triplicate) to the Proportion of CO to GE; and the Proportion of DB to QF, triple to the Proportion of CD to CQ; and consequently, because the Complement BDCFEB is made by the decrease of AC in triple Proportion to that of the decrease of CD, it will be (by the second Article) a third part of the Figure ABEFCA. So that it comes all to one pass, whether we take triple Proporti­on in decreasing to make the Complement, or triple Proportion in increasing to make the Fi­gure; for the Proportion of HF to BA, is triple to the Proportion of CH to CA. Wherefore you have done no more but what you have seen first done, saving that from your construction you prove not the Figure to be triple to the Complement; perhaps because you have proved the con­trary in your Arithmetica Infinitorum. But your way differs from mine, in that you call the Proportion subtriplicate, which I call triplicate; as if the divers naming of the same thing made it differ from its self. You might as well have said briefly, the Proposition is true, but ill proved, because I call the Proportion of one to two triple, or triplicate of that of one to eight; which you say is salse, and hath infected the fourth, fifth, nineth, tenth, eleventh, thir­teenth, fourteenth, fifteenth, sixteenth, seventeenth and nineteenth Articles of the sixteenth Chap­ter. But I say, and you know now, that it is true; and that all those Articles are Demon­strated.

Lastly you add, Tu vero, in presente Articulo, &c. id est, you bid find as many mean Pro­portionals as one will, between two given Lines; as if that could not be done by the Geome­try of Plains, &c. You might have left out Tu vero to seek an Ego quidem. But tell me, do you think you can find two mean Proportionals (which is less then as many as one will) by the Geometry of Plaines? We shall see anon how you go about it. I never said it was im­possible, and if you look upon the places cited by you more attentively, you will find your self mistaken. But I say, the way to do it has not been yet found out, and therefore it may prove a solid Probleme for any thing you know.

The fifth Article you reject, because it citeth the Corollarie of the twenty-eighth Article of the thirteenth Chapter, where there is never a word to that purpose. But there is in the twen­ty-sixth Article; which was my own fault, though you knew not but it might have been the Printers.

To the tenth you object for almost three leaves together, against these words of mine, Because (in the sixth Figure) BF is to BE in triplicate Proportion of CD to FE, therefore inver­ting, FE is to CD in triplicate Proportion of BF to BC. This you objected then. But now that I have taught you so much Geometry, as to know that of three Quantities, begin­ing at the least, if the third be to the first in triplicate Proportion of the second to the first, also by conversion the first to the second shall be in triplicate Proportion of the first to the third; if it were to do again, you would not object it.

My eleventh Article you would allow for demonstrated, if my second had been Deccenstra­ted, upon which it dependeth. Therefore seeing your objections to that Article are sufficiently answered, this Article also is to be allowed.

The twelfth also is allowed upon the same reason. What falsities you shall finde in such fol­lowing Propositions as depend upon the same second Article, we shall then see when I come to the places where you object against them.

[Page 45] To the thirteenth Article you object, That the same Demonstration may be as well applyed to a Portion of any Conoeides Parabolicall, Hyporbolicall, Ellipticall, or any other, as to the Portion of a Sphere. By the truth of this let any man judge of your and my Geometry. Your Comparison of the Sphere and Conocides, so far holds good, as to prove that the Superficies of the Conoeides is greater then the Superficies of the Cone described by the subtense of the Pa­rabolicall, Hyperbolicall, or Ellipticall Line. But when I come to say that The cause of the excess of the Superficies of the Portion of the Sphere above the Superficies of the Cone, con­fisis in the Angle-DAB, and the cause of the excess of the Circle made upon the Tangent AD above the Superficies of the same Cone, consists in the magnitude of the same Angle DAB, how will you apply this to your Con [...]? For suppose that the crocked Line AB (in the se­venth Figure) were not an Arch of a Circle, do you think that the Angles which it maketh with the Subtense AB, at the Points A and B must needs be equall? Or if they be not, does the excess of the Superficies of the Circle upon AD above the Superficies of the Cone, or the exceis of the Superfici [...]s of the Portion of the Conoeides above the Superficies of the same Cone consist in the Angle DAB, o [...] rather in the [...]tude of the two unequall Angles DAB, and ABA? You should have drawn some other crooked Line, and made Tangents to it through A and B, and you would presently have seen your error. See how you can answer this; for if this Demonstration of minestand firm, I may be bold to say, though the same be well Demon­strated by Ar [...], that this way of mine is more naturall, as proceeding immediately from the naturall efficient causes of the effect contained in the conclusion; and besides, more brief and more easie to be followed by the fancy of the Reader.

To the fourteenth Article you say that I commit a Circle in that I require in the fourth Ar­ticle the finding of two mean Proportionals, and come not till now to show how it is to be done. Nor now neither. But in the mean time you commit two mistakes in saying so. The place ci­ted by you in the fourth Article is (in the Latine) Pag. 149. Lin 9. (in the English) Pag. 188. Lin. 3. Let any Reader judge whether that be a requiring it, or a supposing it to be done; this is your first mistake. The second is, that in this place the Preposition itself, which is, If those Deficient Figures could be described in a Parallel [...]gram exquisitely, there might be found there­by between any two Lines given, as many mean Proportionals as one would, is a Theoreme, upon supposition of these crooked Lines exquisitely drawn; but you take it for a Probleme.

And proceeding in that error, you undertake the invention of two mean Proportionals, using therein my first Figure, which is of the same construction with the eighth that belongeth to this fourteenth Article. Your construction is, Let there be taken in the Diameter CA (Figure 1) the two given Lines, or two others Proportionall to them, as CH, CG, and their [...] ­nate Lines HF, GE (which by construction are in subtriplicate Pr [...]p [...]rtion of the intercepted Diameters.) These Lines will shew the Proportions which those four Proportionals are to [...]. But how will you find the Length of HF or GE, the ordinate Lines? Will you not do it by so drawing the crooked Line CFE as it may pa [...]s through both the Points F and E? You may make it pass through one of them, but to make it pass through the other, you must finde two mean Proportionals between GK and GL, or between HI and HP; Which you cannot do, unless the crooked Line be exactly drawn; which it cannot be by the Geometry of Plunes. Go show this Demonstration of yours to Orontius, and see what he will say to it.

I am now come to an end of your objections to the seventeenth Chapter, where you have an Epiphonema not to be passed over in silence. But becaus [...] you p [...]etend to the D [...]tration of some of these Propositions by another Method in your Arithmetica Infinitorum, I shall first try whether you be able to defend those Demonstrations as well as I have done theie of mine by the Method of Motion.

The first Proposition of your Arithmetica Infinitorum is this L [...]mma. In a S [...]ries (or Row) of Quantities Arith [...]tically Proportionall, beginning at a Poi [...]t or Cyp [...]r as 0, 1, 2, 3, 4, &c. to finde the Proportion of the Aggregate of them all, to the Aggregate [Page 46] of so many times the greatest, as there are Terms. This is to be done by multiplying the great­est into half the Number of the Terms. The Demonstration is easie. But how do you de­monstrate the same? The most simple way (say you) of finding this and some other Problemes, it to do the thing it self a little way, and to observe and compare the appearing Proportions, and then by Induction, to conclude it universally. Egregious Logicians and Geometricians, that think an Induction without a Numeration of all the particulars sufficient to infer a Conclusion universall, and fit to be received for a Geometricall Demonstration! But why do you limit it to the naturall consequution of the Numbers, 0, 1, 2, 3, 4, &c? Is it not also true in these Numbers, 0, 2, 4, 6, &c. or in these, 0, 7, 14, 21, &c? Or in any Numbers where the Diff [...]rence of nothing and the first Number is equall to the difference between the first and second, and between the second and third &c? Again, are not these Quantities 1, 3, 5, 7, &c. in continuall Proportion Arith­maticall? And if you put before them a Cypher thus, 0, 1, 3, 5, 7, do you think that the sum of them is equall to the half of five times seven? Therefore though your Lemma be true, and by me (Chap. 13. Art. 5.) demonstrated; yet you did not know why it is true; which also appears most evidently in the first Proposition of your Conique-sections. Where first you have this, That a Parallelogram whose Altitude is infinitely little, that is to say, none, is scarce any thing else but a Line. Is this the Language of Geometry? How do you determine this word scarce? The least Altitude, is Somewhat or Nothing. If Somewhat, then the first character of your Arithmeticall Progression must not be a cypher; and consequently the first eighteen Propositions of this y [...]ur Arithmetica Infinitorum are all naught. If Nothing, then your whole figure is without Altitude, and consequently your Understanding naught. A­gain, in the same Proposition, you say thus, We will sometimes call those Parallelograms ra­ther by the name of Lines then of Parallelograms, at least, when there is no consideration of a determinate Altitude; But where there is a consideration of a determinate Altitude (which will happen sometimes) there that little Altitude shall be so far considered, as that being in­finitely multiplyed it may be equall to the Altitude of the whole Figure. See here in what a con­fusion you are when you resist the truth. When you consider no determinate Altitude (that is, no Quantity of Altitude) then you say your Parallelogram shall be called a Line. But when the Altitude is determined (that is, when it is Quantity) then you will call it a Parallelogram. Is not this the very same doctrine which you so much wonder at and reprehend in me, in your objections to my eighth Chapter, and your word considered used as I used it? 'Tis very ugly in one that so bitterly reprehendeth a doctrine in another, to be driven upon the same himself by the force of truth when he thinks not on't. Again, seeing you admit in any case, those infi­nitely little altitudes to be quantity, what need you this limitation of yours, so far forth as that by multiplication they may be made equall to the Altitude of the whole figure? May not the half, the third, the fourth, or the firth part, &c. be made equall to the whole by multipli­cation? Why could you not have said plainly, so far forth as that every one of those infinitely little Altitudes be not only something but an aliquo [...]part of the whol [...]? [...]o you will have an infi­nitely little Altitude, that is to say a Point, to be both nothing and something and an aliquot part. And all this proceeds from not understanding the ground of your Profession.

Well, the Lemma is true. Let us see the Theoremes you draw from it. The first is (Pag. 3.) that a Triangle to a Parallelogram of equall Base and Altitude is as one to two. The conclu­sion is true, but how know you that? Because (say you) the Triangle consists as it were (as is were, is no Phrase of a Geometrician) of an Infinite Number of stra [...]ght Parallel Lines. Does it so? Then by your own doctrine, which is, that Lines have no breadth, the Altitude of your Triangle consisteth of an infinite Number of no Altitudes, that is of an infinite Number of Nothings, and consequently the Area of your Triangle has no Quantity. If you say that by the Parallels you mean infinitely little Parallelograms, you are never the better; for if infinitely little, either they are nothing, or if somewha [...], yet seeing that no two sides of a Triangle are Parallel, those Parallels cannot be Parallelograms. I see they may be counted for [Page 47] Parallelograms by not considering the Quantity of their Altitudes in the Demonstration. But you are barred of that Plea, by your spightfull arguing against it in your Elenchus. Therefore this third Proposition, and with it the fourth is undemonstrated.

Your fifth Proposition is, The Spirall Line is equall to half the Circle of the first Revoluti­on. But what Spirall Line? We shall understand that by your construction, which is this, The straight Line MA, (in your Figure which I have placed at the end of the fifth Lesson) turned round (the Point M remaining unmoved) is supposed to describe with its Point A the Circle AOA, whilst some Point (in the same MA whilst it goes about) is supposed to be mo­ved uniformly from M to A describing the Spirall Line. This therefore is the Spirall Line of Archimedes; and your Proposition affirms it to be equall to the half of the Circle AOA; which you perceived not long after to be false. But thinking it had been true, you go about to prove it, by inscribing in the Circle an infinite multitude of equall Angles, and consequent­ly an infinite Number of Sectors, whose Arches will therefore be in Arithmeticall Proportion; Which is true. And the Aggregate of those Arches equall to half the Circumference AOA; Which is true also. And thence you conclude that the Spirall Line is equall to half the Circum­ference of the Circle AOA; Which is false. For the Aggregate of that infinite Number of infinitely little Arches, is not the Spirall Line made by your construction, seeing by your con­struction the Line you make is manifestly the Spiral of Archimedes; whereas no Number (though infinite) of Arches of Circles (how little soever) is any kind of Spirall at all; and though you call it a Spirall, that is but a patch to cover your fault, and deceiveth no man but your self. Besides, you saw not how absurd it was (for you that hold a Point to be absolutely nothing) to make an infinite Number of equall Angles (the Radius increasing as the Number of Angles increaseth) and then to say that the Arches of the Sectors whose Angles they are, are as 0, 1, 2, 3, 4, &c. For you make the first Angle 0, and all the rest equall to it; and so make 0, 0, 0, 0, 0, &c. to be the same Progression with 0, 1, 2, 3, 4, &c. The influence of this absurdity reacheth to the end of the eighteenth Proposition. So many are therefore false, or nothing worth. And you needed not to wonder that the Doctrine contained in them was omitted by Archimedes, who never was so senseless as to think a Spi [...]ll Line was compounded of Arches of Circles.

Your nineteenth Proposition is this other Lemma, In a Series (or a Row) of Quantities, beginning from a Point, or Cypher, and proceeding according to the order of the square Numbers, as 0, 1, 4, 9, 16, &c. to finde what Proportion the whole Se­ries hath to so many times the greatest. And you conclude the Proportion to be that of 1 to 3. Which is false, as you shall presently see. First, let the Series of Squares with the prefixed Cypher, and under every one of them, the greatest 4 be [...]And you have for the sum of the Squares 5, and for thrice the greatest 12, the third part whereof is 4. But 5 is greater then 4, by 1, that is, by one twelfth of 12; which Quanti­ty is somewhat, let it be called A. Again, let the Row of Squares be lengthened one term sur­ther, and the greatest set under every one of them as [...] ▪ The sum of the Squares is 14, and the sum of four times the greatest is 36, whereof the third part is 12. But 14 is greater then 12 by two unities, that is, by two twelfths of 12, that is, by 2 A. The difference therefore between the sum of the Squares, and the sum of so many times the greatest Square is greater, when the cypher is followed by three Squares, then when by but two. Again, let the Row have five terms as in these Numbers [...] with the great­est five times subscribed, and the sum of the Squares will be 30, the sum of all the great­est will be 80. The third part whereof is 26 ⅔. But 30 is greater then [Page 48] 26⅓ by 3⅓, that is, by three twelfths of twelve and ⅓ of a twelfth, that is, by 3⅓ A. Likewise in the Series continued to six places with the greatest six times subscribed, as [...] the sum of the Squares is 55, and the sum of the greatest six times ta­ken is 150, the third part where [...]f is 50. But 55 is greater then 50 by 5, that is, by five twelfths of 12, that is, by 5. A. And so continually as the Row groweth longer, the excess also of the aggre­gate of the Squares above the third part of the aggregate of so many times the greatest Square, grow­eth greater. And consequently if the Number of the Squares were infinite, their sum would be so far from being equal to the third part of the aggregate of the greatest as often taken, as that it would be greater then it by a Quantity greater then any that can be given or named.

That which deceived you was partly this, that you think (as you do in your Elenchus) that these Fractions [...] &c. are Proportions, as if 1/12 were the Proportion of one to twelve, and consequently 2/12 double the Proportion of one to twelve; which is as un­intelligible as School-Divinity; and I assure you, far from the meaning of Mr. Ougthred in the sixth Chapter of his Clavis Mathematicae, where he sayes that 43/7 is the Proportion of 31 to 7; for his meaning is, that the Proportion of 43/7 to one, is the Proportion of 31 to 7; whereas if he meant as you do, then 86/7 should be double the Proportion of 31 to 7. Partly also because you think (as in the end of the twentieth Proposition) that if the Proportion of the Numerators of these Fractions [...] to their Denominators decrease eternally, they shall so vanish at last as to leave the Proportion of the sum of all the Squares to the sum of the greatest so often taken, (that is, an infinite Number of times) as one to three, or the sum of the greatest to the sum of the increasing Squares, as three to one; for which there is no more reason then for four to one, or five to one, or any other such Proportion. For if the Proportions come eternally nearer and nearer to the subtriple, they must needs also come nearer and nearer to subqua­druple; and you may as well conclude thence that the upper Quantities shall be to the Lower Quantities as one to four, or as one to five, &c. as conclude they are as one to three. You can see without admonition, what effect this false ground of yours will produce in the whole struc­ture of your Arithmetica Infinitorum; and how it makes all that you have said unto the end of your thirty-eighth Proposition, undemonstrated, and much of it false.

The thirty-nineth is this other Lemma, In a Series of Quantities beginning with a Point or Cypher, and proceeding according to the Series of the Cubique Numbers, as 0. 1. 8. 27. 64, &c. to finde the Proportion of the sum of the Cubes to the sum of the greatest Cube, so ma­ny times taken as there be Terms, And you conclude that they have the Proportion of 1 to 4; which is false.

Let the first Series be of three terms subscribed with the greatest [...]; the sum of the Cubes is nine; the sum of all the greatest is 24; a quarter whereof is 6. But 9 is greater then 6 by three unities. An unity is something. Let it be therefore A. Therefore the Row of Cubes is greater then a quarter of three times eight, by three A. Again, let the Series have four terms, as [...]; the sum of the Cubes is 36; a quarter of the [Page 49] sum of all the greatest is twenty-seven. But thirty-six is greater then twenty-seven by nine [...] that is, by 9 A. The excess therefore of the sum of the Cubes above the fourth part of the sum of all the greatest, is increased by the increase of the Number of terms. Again, let the terms be five as [...] the sum of the Cubes is one hundred; the sum of all the greatest three hundred and twenty; a quarter whereof is eighty. But one hundred is greater then eighty by twenty, that is, by 20 A. So you see that this Lemma also is false. And yet there is grounded upon it all that which you have of comparing Parabolas and Paraboloeides with the Parallelograms wherein they are accommodated. And therefore though it be true, that the Parabola is ⅔, and the Cubicall Paraboloeides ¾ of their Parallelograms respectively' yet it is more then you were certain of when you referred (me for the learning of Geometry) to this Book of yours. Besides, any man may perceive that without these two Lemmas (which are mingled with all your compounded Series with their excesses) there is nothing demonstra­ted to the end of your Book. Which to prosecute particularly, were but a vain expence of time. Truly were it not that I must defend my reputation, I should not have shewed the world how little there is of sound Doctrine in any of your Books. For when I think how dejected you will be for the future, and how the grief of so much time irrecoverably lost, together with the conscience of taking so great a stipend, for mis-teaching the young men of the University, & the consideration of how much your friends wil be ashamed of you, will accompany you for the rest of your life, I have more compassion for you then you have deserved. Your Treatise of the Angle of Contact, I have before confuted in a very few leaves. And for that of your Conique Sections, it is so covered over with the scab of Symboles that I had not the patience to examine whether it be well or ill demonstrated.

Yet I observed thus much, that you find a Tangent to a Point given in the Section, by a Diameter given; and in the next Chapter after, you teach the finding of a Diameter, which is not artificially done.

I observe also, that you call the Parameter an Imaginary Line, as if the place thereof were less determined then the Diameter it self; and then you take a mean Proportionall between the intercepted Diameter, and its contiguous ordinate Line to find it. And tis true, you find it [...] But the Parameter has a determined Quantity to be found without taking a mean Proportional. For the Diameter and half the Section being given, draw a Tangent through the Vertex, and dividing the Angle in the midst which is made by the Diameter and Tangent, the Line that so divideth the Angle, will cut the crooked Line. From [...] intersection draw a Line (if it be a Parabola) Parallel to the Diameter, and that Line shall cut off in the Tangent from the Vertex the Parameter sought. But if the Section be an E [...]lipsis, or an Hyperbole, you may use the same Method, saving that the Line drawn from the intersection must not be Parallel, but must pass through the end of the transverse Diameter, and then also it shall cut off a part of the Tan­gent, which measured from the Vertex is the Parameter. So that there is no more reason to call the Parameter an Imaginary Line then the Diameter.

Lastly, I observe that in all this your new Method of Coniques you shew not how to find the Burning Points, which writers call the Foci and Umbilici of the Section, which are of all other things belonging to the Coniques most usefull in Philosophy. Why therefore were they not as worthy of your pains as the rest, for the rest also have already been demonstrated by others? You know the Focus of the Parabola is in the Axis distant from the Vertex a quarter of the Pa­rameter. Know also that the Focus of an Hyperbole, is in the Axis, distant from the Vertex, as much as the Hypotenusall of a rectangled Triangle, whose one side is half the transverse Axis, the other side half the mean Proportionall between the whole transverse Axis and the Pa­rameter, is greater then half the transverse Axis.

The cause why you have performed nothing in any of your Books (saving that in your E­len [...]hus you have spied a few negligences of mine, which I need not be ashamed of) is this, that [Page 50] you understood not what is Quantity; Line, Super [...]ies, Angle, and Proportion; without which you cannot have the Science of any one Proposition in Geometry. From this one and first Definition of Euclise, a Point is th [...] whereof there is no p [...]t, understood by Sextus Empiricus, as you understand it, that is to say, mis-understood, Sextus Empiricus hath utterly destroyed most of the rest, and Demonstrated, that in Geometry there is no Science, and by that means you have betrayed the most evident of the Sciences to the Sceptiques. But as I under­stand it for that whereof no part is reckoned, his Arguments have no force at all, and Geometry is redeemed. If a Line have no Latitude, how shall a Cylinder rowling on a Plain, which it toucheth not but in a Line, describe a Superficies? How can you affirm that any of those things can be without Quantity, whereof the one may be greater or less then the other? But in the common Contact of divers Circles the externall Circle maketh with the common Tan­gent a less Angle of Contact then the internall. Why then is it not Quantity? An Angle is made by the concourse of two Lines from severall Regions, concurring (by their generation) in one and the same Point. How then can you say the Angle of Contact is no Angle? One measure cannot be applicable at once to the Angle of Contact, and Angle of Conversion. How then can you infer, if they be both Angles, that they must be Homogeneous? Proportion is the Relation of two Quantities. How then can a Quotient or Fraction, which is Quanti­ty absolute, be a Proportion? But to come at last to your Ephiphonema, wherein, though I have perfectly demonstrated all those Propositions concerning the Proportion of Parabolasters to their Parallelog [...]ams, and you have demonstrated none of them (as you cannot now but plainly see) but committed most [...] Paralogisms, How could you be so transported with pride, as insolently to compare the setting of them forth as mine, to the Act of him that steals a horse, and comes to the gallows for it. You have read, I think, of the gallows set up by Ha­man. Remember therefore also who was hanged upon it.

After your dejection I shall comfort you a little, a very little, with this, that whereas this 18 Chapter containeth two Problems, one, the finding of a straight Line equall to the [...]rook­ed Line of a Semiparabola, The other, the finding of straight Lines equall to the crocked Lines of the Parabolasters in the table of the third Article of the 17 Chapter; You have [...]u­ly demonstrated that they are both false; and another hath also Demonstrated the same another way. Nevertheless the fault was not in my method, but in a mistake of one Line for another, and such as was not hard to correct; and is now so corrected in the English as you shall not be able (if you can sufficiently imagine Motions) to reprehend. The fault was this, That in the Triangles which have the same. Base and Altitude with the Parabola and Parabo­laster, I take for designation of the mean uniform Impetus, a mean Proportionall (in the first Figure) between the whole Diameter and its half, and (in the second Figure) a mean Proporti­onall between the whole Diameter and its third part; which was manifestly false, and contrary to what I had shewn in the 16 Chapter. Whereas I ought to have taken the half of the Base, as now I have done, and thereby exhibited the Straight Lines equall to those crocked Lines, as I undertook to do. Which error therefore proceeded not from want of skill, but from want of care; and what I promised (as bold as you say the promise was,) I have now performed.

The rest of your exceptions to this Chapter, are to these words in the end, there be some that say, that though there be equality between a straight and crooked Line, yet now, (they say) after the full of Adam, it cannot be found without the especiall help of divine grace. And you say you think there be none that say so. I am not bound to tell you who they are. Never­theless, that other men may see the Spirit of an ambitious part of the Clergy, I will tell you where I read it. It is in the Prolegomena of Lalovera (a Jesuite) to his Quadrature of the Cir­cle, pag. 13 & 14, in these wor [...]s, Quamvis circuli tetragonismus fit [...] possibilis, an taman etiam sit [...], hoc est, post Adae lapsum homo ejus scientiam abs (que) speciali di­ [...]inae gratiae auxilio, possit comparare, jure merito inquirunt Theologi, pronunciant (que) hanc veritatem tanta esse caligine involutam ut illam videre nemo possit, nisi Ignorantiae ex pri­mi [Page 51] parentis praevaricatione propagatas tenebras indebitus divina lucis radius dissipet; quod ve­rissimum esse sentio. Wherein I observed that he (supposing he had found that Quadrature) would have us believe it was not by the ordinary and Naturall help of God (whereby one man reasoneth, judgeth and remembreth better then another) but by a Special (which must be a Super­turall) help of God, that he hath given to him of the order of Jesus above others that have attempted the same in vain. Insinuating thereby, as handsomely as he could, a Speciall love of God towards the Jesuites. But you taking no notice of the word Speciall, would have men think I held, that humane Sciences might be acquired without any help of God. And there­upon proceed in a great deal of ill language to the end of your objections to this Chapter. But I shall take notice of your Manners for altogether in my next Lesson.

At the nineteenth Chapter you see not (you say) the Method. Like enough. In this Chap­ter I consider not the Cause of Reflection, which consisteth in the resistance of Bodies naturall; but I consider the consequences, arising from the supposition of the equality of the Angle of Reflection, to that of Incidence; leaving the causes both of Reflection, and of Refraction to be handled together in the 24 Chapter. Which Method (think what you will) I still think best.

Secondly, you say I define not here, but many Chapters after, what an Angle of Incidence, and what an Angle of Reflection is. Had you not been more hasty then diligent Readers, you had found that those Definitions of the Angle of Incidence, and of Reflection were here set down in the first Article, and not deferred to the 24. Let not therefore your own oversight be any more brought in for an objection.

Thirdly, you say there is no great difficulty in the business of this Chapter. It may be so, now 'tis down; but before it was done, I doubt not but you that are a Professor would have done the same, as well as you have done that of the Angle of Contact, or the business of your Arithmetica Infinitorum. But what a novice in Geometry would have done I can­not tell.

To the third, fourth, and fifth Article, you object a want of Determination; and shew it by instance, as to the third Article. But what those Determinations should be, you determine not, because you could not. The words in the third Article, are first these, If there fall two straight Lines Parallel, &c. which is too generall. It should be, if there fall the same way two straight Lines Parallel, &c. Next these, their Reflected Lines produced inwards shall make an Angle, &c. This also is too generall. I should have said their Reflected Lines produced inwards, if they meet within, shall make an Angle, &c. Which done, both this Article and the 4 and 5 are fully demonstrated. And without it, an intelligent Reader had been satisfied, supplying the want himself by the construction.

To the eight, you object onely the two great Length, and labour of it, because you can do it a shorter way. Perhaps so now, as being easie to shorten many of the Demonstrations both of Euclide, and other the best Geometricians that are or have been. And this is all you had to say to my 19 Chapter. Before I proceed, I must put you in mind that these words of yours, Adducis malleum, ut occi [...]as muscam, are not good Latine, Malleum affers, Malleum adhi­ [...]es, Malleo uteris, are good. When you speak of bringing Bodies animate, Ducere and Adduc [...] ­re are good, for there to bring, is to gu [...]e or lea [...]. And of Bodies inanimate Adducere is good for Attra [...]ere, which is to draw to. But when you bring a hammer, will you say Adduco Malleum, I lead a hammer? A man may lead another man, and a ninny may be said to lead another ninny, but not a hammer. Neverthelese, I should not have thought fit to reprehend this fault upon this occasion in an English man, nor to take notice of it, but that I finde you in some places nibling (but causelessly) at my Latine.

Concerning the twentieth Chapter, before I answer to the objections against the Propositions themselves, I must answer to the exception you first take to these words of mine, Quae de di­mensione Circuli & Angulorum pronuntiata sunt tanquam exacté inventa, accipiat Lector [...]n­quam dicta Problematice. To which you say thus, we are wont in Geometry to call some Pro­positions [Page 52] Theorem [...]s, other's Problems, &c. of which [...] Theoreme is that wherein some asser­tion is propou [...]nded to be proved, a Problem that wherein something is commanded to be done. Do you mean to be done, and not proved? By your favour, a Probleme in all ancient writers signifies no more but a Proposition uttered, to the end to have it by them to whom it is uttered, examined whether it be true or not true, fai [...]able or not faisable; and differs not amongst Geo­metricians from a Theoreme, but in the manner of Propounding. For this Proposition, To make an equilaterall Triangle, so propounded they call a Problem. But if propounded thus, If upon the ends of a straight Line given be described two Circles, whose Radius is the same straight Line, and there be drawn from the intersection of the Circles to their two Centers, two straight Lines, there will be made an equilateral Triangle, then they call it a Theoreme; and yet the Proposition is the same. Therefore these words Accipiat Lector tanquam dicta Pro­blemati [...]ey, signifie plainly this, that I would have the Reader, take for p [...]opounded to him to examine, whether from my construction the Quadrature of the Circle can be truly inferred or not; and this is not to bid him (as you inte [...]pret it) to square the Circle. And if you beli [...]ve that Problematicey signifies probably, you have been very negligent in observing the sense of the an [...]ien [...] G [...]eek Philosophers in the word Probleme. Therefore your Solemus in Geometria, &c. is nothing to the pu pose; nor had it been though you had spoken more properly, and said So­l [...]nt, leaving out your selves.

My first Article hath this Title, from a False supp [...]sition, a false Quadrature of the Circle. Seeing therefore you were resolved to shew where I erred, you should have proved either that the Supposition was true, and the Conclusion falsely inferred, or contrarily that though the Suppo­sition be false, yet the Conclusion is true; for else you object nothing to my Geometry, but only to my Judgement, in thinking fit to publish it; which nevertheless you cannot justly do, seeing it was likely to give occasion to ingenuous men (they practise of it being so accurate to sense) to inquire wherein the Fallacy did consist. And for the Probleme as it was first printed, but never published, and consequently ought to have passed for a private paper stoln out of my study, your publique objecting against it, (in the opinion of all men that have conversed so much with honest company as to know what belongs to civill conversation,) was sufficiently bar­barous in Divines. And seeing you knew I had rejected that Proposition, it was but a poor Am­bition to take wing as you thought to do, like Beetles from my egestions. But let that be as it will, you will think strange now I should resume, and make good (at least against your objection) that very same Proposition. So much of the Figure as is needfull you will finde noted with the same letters, and placed at the end of this 5 Lesson. Wherein let B I, be an Arch not greater then the Radius of the Circle, and divided into four equall parts, in L, N, O. Draw S N, the Sine of the Arch B N, and produce it to T, so as S T be double to S N, that is, equall to the Chord B I. Draw likewise a L, the Sine of the Arch B L, and produce it to c, so as a c be qua­druple to a L, that is, equall to the two Chords B N, N I. Upon the Center N with the Ra­dius N I, draw the A [...]ch I d, cutting B U the Tangent in d. Then will B N produced cut the Arch I d, in the midst at o. In the Line B S produced take S b, equall to B S; then draw and produce b N, and it will fall on the Point d. And B d, S T, will be equall; and d T joyned and produced will fall upon o, the midst of the Arch I d. Joyn I T, and produce it to the Tangent B U in U. I say, that the st [...]aight Line I T U shall pass through c. For seeing B S, S h, are equall, and the Angle at S a right Angle, the straight Lines B N, and b N, are also equall, and the Triangles B N b, d N o like and equall; and the Lines d T, T o equall. Draw o i Parallel to d U, cutting I U in i; and the Triangles d T v, o T i will also be like and equal. Produce S T to the Arch d o I in e, and produce it further to f, so that the Line e f be equall to T c; and then S f will be equall to a c. Therefore f c joyned will be Parallel to B S. In c f produced take f g equall to c f; and draw g m Parallel to d U, cutting I U in m, and d o in n; and let the inte [...]ction of the two Lines a c and d o be in r; which being done, the Triangles m n T, r c T will be like and equall. Therefore m n and r c are equall; and [...]sequently the st [...]aight Line I m T U shall pass through c. Dividing therefore a c in the midst at t, and S N [Page 53] in the midst at l, and joyning t N, L l, the Lines L l, t N, and c T produced with all meet i [...] one and the same Point of B S produced; suppose at q. Therefore the Point q being given by the two known Points T and I, the Lines drawn from q through equall parts of the Sine of the Arch B I, for example through the Points P, Q, R, of the Sine M I, shall cut off equall Ar­ches, as B L, L N, N O, O I. And this is enough to make good that Probleme, as to your objection.

The straight Line therefore B U for any thing you have said is p [...]oved equall to the Arch B I, and the division of any Angle given into any proportion given, the Quadrature of any Sector, and the Construction of any equilaterall P [...]lygon is also given. And though in this also I should have erred, yet it cannot be denyed but that I have used a more natural, a more Geometrical, and a more pe [...]spicuous method in thes search of this so difficult a Probleme, then you have done in your A­rithmetica Infinitorum For though it be true that the aggregate of all the mean Proportionals be­tween the Radius together with an infinitely little part of the same, and the Radius wanting an in­finitely little part of the same; and again, between the Radius, together with two infinitely little parts, and the Radius wanting two infinitely little parts, and so on eternally will be equall to the Quadrant (a thing which every mean Geometrician knew before) yet it was absurd to think those Means could be calculated in Numbers by Interpoling of a Symbole; especially when you make that Symbole to stand for a Numbet neither true nor surd; as if there were a number that could neither be uttered in words, nor not be uttered in words. For what else is surd, but that which cannot be spoken?

To the fifth Article (though your discourse be long) you object but two things. One is, that Whereas the Spirall of A chi [...]edes is made of two Motions, one straight, the other cir­cular, both uniform, I taking the Motion compounded of them both for one of those that are compounded, conclude falsely, that the generation of the Spirall is like to the generation of the Parabola. What heed you use to take in your rep [...]ehensions, appears most manifestly in this objection. For I say in that demonstration of mine that the velocity of the point A in descri­bing the Spirall, en [...]reaseth continually in Proportion to the Times. For seeing it goes on uniformly in the Semidiameter, it is impossible it should not pass into greater and greater Cir­cles proportionally to the Times; and consequently it must have a swifter and swifter Motion circular, to be compounded with the uniform Motion in every Point of the Radius as it turn­eth about. This objection therefore is nothing but an effect of a Will (without cause) to con­tradict.

The other objection is that Granting all to be true hith [...]rto, yet because it depends upon the finding of a straight Line equall to a Parabolic [...]ll Line in the 18 Chapter where I was deceiv­ed, I am also deceived here. True. But because in the 18 Chapter of this English E [...]ition I have found a straight Line equall to a Parabolicall Line, I have also found a straight Line equall to the Spirall Line of Archime [...]s. I must here p [...]t you in minde that by these words in your objections to the fifth Article at your Number 2, Quatenus verum [...]st, &c. we have demonstrated Prop. 10, 11, 13. Arithmet. Infinit. you make it appear that you thought your Spirall (made of A [...]ches or Circles) was the true Spirall of Archimedes; which is fully [...]s ab­surd as the Quadrature of Joseph Scaliger, whose Geometry you so much d [...]spise.

To the sixth Article which is a Digression concerning the Analytiques of Geomet [...]ici­a [...]s, you deny that the Efficient cause of the Construction ought to be contained in the De­monstration. As if any Probleme could be known to be truly done, otherwise then by knowing first how, that is to say, by what Efficient Cause, and in what manner it is to be done. What­soever is done without that knowledge, cannot be demonstrated to be done; as you see in your computation of the Parabola, and Parabolocides, in your Arithmetica Infini­torum.

And whereas I said that The ends of all straight Lines drawn from a straight Line, and passing through one and the same Point, if their parts be Proportionall, shall be in a [...]aight Line; is true and accurate; as also If they begin in the Circumference of a Cir [...]le, they [...] [Page 54] also be in the Cir [...]umference of another Circle. And so is this, If the Proportion be dupli­cate, they shall be in a Parabola. All this I say is true and accurately spoken. But this was no place for the demonstration of it. Others have done it. And I perceive by that you put in by Parenthesis (intelligi [...] credo inter [...]du [...]s Peripheri [...] concentric [...]s) that you understand not what I mean.

Hitherto reach your objections to my Geometry. For the rest of your Book, it contain­eth nothing but a collection of lies, wherewith you do what you can, to extenuate as vulgar, and disgrace as false, that which followeth, and to which you have made no speciall ob­jection.

I shall therefore only add in this place concerning your Analytica per Potestates, that it is no Art. For the Rule, both in Mr. Ougthred, and in Des Cartes is this, When a Probleme or Question is propounded, suppose the thing required done, and then using a fit ratiocination, put A or some other vowell for the magnitude sought. How is a man the better for this Rule without another rule, How to know when the ratiocinatión is fit? There may therefore be in this kind of Analysis more or less naturall prudence, according as the Analyst is more or less wis [...], or as one man in chusing of the unknown Quantity with which he will begin, or in chu­sing the way of the consequences which he will draw from the Hypothesis, may have better luck then another. But this is nothing to Art. A man may sometimes spend a whole day in deri­ving of consequences in vain, and perhaps another time solve the same Probleme in a few minutes.

I shall also add, that Symboles though they shorten the writing, yet they do not make the Reader understand it soon [...]r then if it were written in words. For the conception of the Lines and Figures (without which a man learneth nothing) must proceed from words, either spoken or thought upon. So that there is a double labour of the mind, one to reduce your Symboles to words (which are also Symboles) another to attend to the Ideas which they signifie. Besides, if you but consider how none of the Antients ever used any of them in their published demon­strations of Geometry, nor in their Books of Arithmetique, more then for the Rootes and Potestates themselves; and how bad success you have had your self in the unskilfull using of them, you will not, I think, for the future be so much in love with them as to demonstrate by them that first part you promise of your Opera Mathe [...]atica. In which if you make not amends for that which you have already published, you will much disgrace those Mathematici­ans you address your Epistles to, or otherwise have commended; as also the Universities (as to this kinde of Learning) in the sight of learned men beyond Sea. And thus having examined your panier of Mathematiques, and finding in it no knowledge neither of Quantity, nor of measure, nor of Proportion, nor of Time, nor of Motion, nor of any thing, but only of certain Characters, as if a Hen had been scraping there; I take out my hand again, to put it in to your other panier of Theology, and good Manners. In the mean time I will trust the objections made by you the Astronomer (wherein there is neither close reasoning, nor good stile, nor sharpness of wit, to impose upon any man) to the discretion of all sorts of Readers.



Of MANNERS. To the same egregious Professors of the Ma­thematicks in the University of Oxford.

HAving in the precedent Lessons maintained the Truth of my Geometry, and sufficient­ly made appear, that your objections against it are but so many errors of your own, proceeding from misunderstanding of the Porpositions you have read in Euclide, and other Masters of Geometry; I leave it to your consideration to whom belong (according to your own sentence) the unhandsome attributes you so often give me upon supposition, that you your selves are in the right, and I mistaken; and come now to purge my self of those greater accusations which concern my Manners. It cannot be expected there should be much Science of any kinde in a man that wanteth Judgement; nor Judgement in a man that knoweth not the Manners due to a publique disputation in writing; wherein the scope of either party ought to be no other then the examination and manifestation of the truth. For whatsoever is added of contumely, ei [...]er directly, or scommatically, is want of Charity, and uncivil; unless it be done by way of Reddition from him that is first provoked to it. I say unless it be by way of Reddition; for so was the Judgement given by the Emperor Vespasian in a quarrell between a Senato: and a Knight of Rome which had given him ill language. For when the Knight had proved, that the first ill language proceeded from the Senator, the Emperor acquitted him in these words Maledici senctor ibus non oportere; remal [...]dicere, fas & civtle esse. Neverthe­less, now a dayes uncivill words are commonly and bitterly used by all that write in matter of Controversie, especially in Divinity, excepting now and then such writers as have been more then ordinarily well bred, and have observed, how hainous, and ha [...]ardous a thing such c [...]n­tumely is amongst some sorts of men, whether that which is said in disgrace be true or false. For evill words by all men of understanding are taken for a defiance, and a challenge to open war. But that you should have bserved so much, who are yet in your mothers belly, was not a thing to be much expected.

The faults in Manners you lay to my charge, are these, 1. Self conceit. 2. That I will be very angry with all men that do not presently submit to my Dictates. 3. That I had my Do­ctrine concerning Vision, out of papers which I had in my hands of Mr. Warners. 4. That I have injured the Universities. 5. That I am an Enemy to Religion. These are great faults; but such as I cannot yet confess. And therefore I must as well as I can, seek out the grounds up­on which you build your Accusation. Which grounds (seeing you are not acquainted with my conve [...]sation) must be either in my published writings, or reported to you by honest men, and without suspition of interest in reporting it. As for my self-conceit and ostentation, you shall finde no such matter in my writings That which you alleadge from thence is first that in the Epistle Dedicatory I say of my Book de Corpore, Though it be little, yet it is full; and if good may go for great, great enough. When a man presenting a gift great or small to his [Page 56] betters, adorneth it the best he can to make it the more acceptable; he that thinks this to be Ostentation, and self-conceit, is little versed in the common actions of humane life. And in the same Epistle where I say of Civill Philosophy, it is no antienter then my Book de Cive, these words are added, I say it provoked, and that my detractors may see they lose their labour. But that which is truly said, and upon provocation, is not boasting, but defence. A short sum of that Book of mine, now publiquely in French, done by a Gentleman I never saw, carrieth the Title of Ethiques demonstrated. The Book it self translated into French hath not onely a great testimony from the Translator Serberius, but also from Gassendus, and Mersennus, who being both of the Roman Religion had no cause to praise it, or the Divines of England have no cause to finde fault with it. Besides, you know that the Doctrine therein contained is gene­rally received by all but those of the Clergy, who think their interest concerned in being made subordinate to the Civil Powe [...]; whose testimonies therefore are invalide. Why there­fore if I commend it also against them that dispraise it publiquely, do you call it boasting? You have heard (you say) that I had promised the Quadrature of the Circle, &c. You heard then that which was not true. I have been asked sometimes, by such as saw the Figure before me, what I was doing, and I was not a [...]aid to say I was seeking for the solution of that Pro­bleme; but not that I had done it. And afterwards being asked of the success, I have said, I thought it done. This is not boasting; and yet it was enough, when told again, to make a fool believe 'twas boasting. But you the Astronomer in the Epistle before your Philosophicall Essay, say you had a great expectation of my Philosophicall, and Mathematicall works; be­fore they were published. It may be so. Is that my fault? can a man raise a great expectati­on of himself by boasting? If he could, neither of you would be long before you raised it of your selves; saving that what you have already published, has made it now too late. For I verily believe there was never seen worse reasoning then in that Philo [...]ophicall Essay; which any judicious Reader would believe proceeded from a Praevaricator, rather then from a man that believed himself; nor worse Principles, then those in your Books of Geometry. The ex­pectation of that which should be written by me, was raised partly by the Cogitata Physico-Mathematica of Mersennus, wherein I am often named with honour; and partly by others with whom I then conversed in Paris, without any ostentation. That no man has a great expectation of any thing that shall proceed from either of you two, I am content to let it be your praise.

Another Argument of my self-conceit, you take from my contempt of the writers of Me­taphysiques and School-Divinity. If that be a sign of self-conceit, I must confess I am guilty; and if your Geometry had then been published, I had contemned that as much. But yet I cannot see the consequence (unless you lend me your better Logique) from despising insignifi­cant and absurd language to self-conceit.

And again, in your Vindiciae Academiarum, you put for boasting, that in my Leviathan Pag. 180. I would have that Book by entire Soveraignty imposed upon the Universities; and in my Review Pag. 395. That I say of my Leviathan, I think it m [...]y be profitably printed, and more profitably taught in the University. The cause of my writing that Book, was the consideration of what the Ministers before, and in the beginning of the Civill War, by their preaching and writing did contribute thereunto. Which I saw not onely to [...]end to the Abate­ment of the then Civill Power, but also to the gaining of as much thereof as they could (as did ofterwards more plainly appear) unto themselves. I saw also that those Ministers, and many other Gentlemen who were of their Opinion, brought their Doctrines against the Civill Power from their Studies in the Universities. Seeing therefore that so much as could be contributed to the Peace of our Country, and the settlement of Soveraign Power without any Army, must proceed from Teaching; I had reason to wish, that Civill Doctrine were truly taught in the Universities. And if I had not thought that mine was such, I had never written it. And ha­ving written it, if I had not recommended it to such as had the Power to cause it to be taught, [Page 57] I had written it to no purpose. To me therefore that never did write any thing in Philosophy to show my Wit, but (as I thought at least) to benefit some part or other of mankind, it was very necessary to commend my Doctrine to such men as should have the Power and Right to Regulate the Universities. I say my Doctrine; I say not my Leviathan. For wiser men may so digest the same Doctrine as to fit it better for a publique teaching. But as it is, I beli [...]ve it hath framed the mines of a thousand Gentlemen to a consciencious obedience to present Go­vernment, which otherwise would have wave [...]ed in that Point. This therefore was no vanting, but a necessary part of the business I took in hand. You ought also to have considered, that this was said in the close of that part of my Book which concerneth Policy meerly Civill. Which part if you the Astronomer, that no [...] think the Doctrine unworthy to be taught, were pleased once to honour with praises printed before it, you are not very constant, no [...] ingenuous. But whether you did so or not, I am not curtain, though it was [...]old me for certain. If it were not you, it was some body else whose Judgement has as much weight at least as yours.

And for any thing you have to say from your own knowledge, I remember not that ever I saw eithe [...] of your faces. Yet you the Professor of Geometry go about obliquely to make me be­lieve that Vindex hath discoursed with me, once at least, though I remember it not. I sup­pose it therefore true; But this I am sure is false, that either he or any man living did ever hear me brag of my Science, or praise my self, but when my defence required it. Perhaps some of our Philosophe [...]s that were at Paris at the same time, and acquainted with the same Learned men that I was acquainted with, might take for bragging the maintaining of my opinions, and the not yiel [...]ing to the reas [...]ns alieadged against them. If that be ostentation, they tell you the truth. But you that are so wise should have considered, that even such men as profess Philoso­phy are carried away with the passions of Emulation and Envy (the sole ground of this your accusation) as well as other men, and instanced in your selves. And this is sufficient to shake off your aspersions [...]f Ostentation and Self-conceit. For if I added, that my acquaintance know that I a [...] naturally of modest rather then of boasting speech, you will not believe it, becaus [...] you distinguish not between that which is said upon provocation, and that which is said without [...], from vain glory.

The next accusation is, That I will be very angry with all men that do not presently submit to my dictates; and that for advancing the reputation of my own skill, I care not what un­wo [...]y [...] [...]ast on others. This is in the Epistle placed before the Vindiciae Academia­rum, subscribed by N S, as the Plain Song for H D in the rest of the Book to D [...]scant upon. I know well enough the Auth [...]s names; and am sorry that N S has lent his name to be abu­sed to so ill a p [...]pos [...]. But how does this appear? What Argument, what witness is t [...]ere of it? You offer no [...]e; or a [...] I conscious of any. I begin to suspect, since you the P [...]ofessor of Geo­metry have in your objections to the 20 Chapter these words concerning Vindex, O [...]ularis il­le testis de quo h [...]c agitur, erat, ni fallor, ille ipse; That Vindex himself, in other company, has [...] a visit on me. Seeing you will have me believe it, let it be so; and (as it is likely) not long after my return into England. At which time (for the [...]putation, it seems, I had gotten by my boasting) divers persons that professed to love Philosophy and Mathematiques, came to s [...]e me; and some of [...] to let me see them, and hear and applaud what they applauded in t [...]mselves. I see now it hath happened to me with Vin [...]ex, as it happened to Dr. Harvey with Mor [...]s. M [...]ranus [...] jesui [...]e came out of Flanders hither, especially (as he sayes) to see what lea [...]ned men in. Divinity, E [...]i [...]ues, Physiques, and G [...]ometry were here yet alive, to the end [...] by di [...]coursing with th [...]m in these Sciences, he might correct either his own, or their [...], [...] [...] [...] was b [...]ought ( [...] say [...]s) to that most civill and renowned old man Dr. Ha [...]vey. [...] very well. And in good ea [...]nest if he had made good use of the Time which was v [...]y [...] afforded him, he might have learned of him (or of no man living) very [...] k [...]ow­ledge conce [...]ning the Circulation of the blood, the Generation of livin [...] Creatures, a [...]d m [...]ny [Page 58] other difficult Points of naturall Philosophy. And if he had had any thing in him but com­mon and childish learning, he could have shewed it no where more to his advantage, then be­fore him that was so great a Judge of such matters. But what did he? That pretious time (which was bat little, because he was to depart again presently for Flanders) he bestowed wholly in venting his own childish Opinions, not suffering the Doctor scarce to speak; losing thereby the benefit he came for, and discovering that he came not to hear what others could say, but to show to others how learned he was himself already. Why else did he take so little time, and so mispend it? Or why returned he not again? But when he had talked away his time, and found (though patiently and civilly heard) he was not much admired, he took occa­sion (writing against me) to be revenged of D. Harvey, by sleighting his learning publiquely; and tels me that his learning was onely Experiments, which he sayes I say have no more certainty then Civil Histories. Which is false. My words are, Ante hos nihil certi in Physicâ erat praeter Experimenta [...]ui (que) sua, & Historias Naturales, si tamen & [...]ae dicendae [...]ertae sint, quae Civilibus Historiis ce [...]tiores n [...] sun [...]. Where I except expressly nom uncertainty the Expe­riments that every man maketh to himself. But you see the [...]ere-cut, by which vain Glory joyn­ed with Ignorance passeth quickly over to E [...]vy and Contumely.

Thus it seems by your own confession I was used by Uin [...]x. He comes with some of my ac­quaintance in a Visit. What he said I know not, but if he [...]iscou [...]sed then, as in his Philoso­phicall Essay he writeth, I will be bold to say of my self, I was so far from morosity, o [...] (to use his Phrase) from being tetricall, as I may very well have a good opinion of my own patience. And if there passed between us the discourse you mention in your Elenchus, Pag. 116. it was an incivility in him so great, that without great civility I could not have abstained from bidding him be gone. That which passed between us, you say was this. I complained that whereas I made Sense, nothing but a perception of Motion in the Organ, nevertheless the Philosophy Schools through all Europel [...]d by the Text of A istotle, teach another Doctrin, namely, that Sen­sation is performed by Species. This is a little mistaken. For I do glory, not complain, that whereas all the Universities of Europe hold Sensation to proceede from Species, I [...] it to be a perception of Motion in the Organ. The answer of Vindex, you say, was, That the other Hy­pothesis, whereby Sense was explicated by the Principles of Motion, was commonly admitted here before my Book came out, as having been sufficiently delivered by Des Car [...]s, Gassendus and Sir Kenelme Digby before I had published any thing in this kinde. This then, it seems, was it that made me angry. Truly I remember not any angry word that ever I uttered in all my life to any man that came to see me, though some of them have troubled me with very imperti­nent discourse; and with those that argued with me, how [...]pertinently s [...]ever, I alwayes thought it more civility to be somewhat earnest in the defence of my opinion, then by obstinate and affected silence to let them see I contemned them, or hea [...]kned not to what they said. I [...] I were earnest in making good, that the manner of Sensation by such Motion as I had explicated in my Leviathan, is in none o [...] the Authors by him named, it was not Anger, but a ca [...]e of not offending him, with any signe of the contempt [...]hich his discourse deserved. But it was Inci­vility in him to make use of a Visi [...], which all men take for a p [...]ofessi [...]n of Friendship, to tell me that that which I had already published for my own, was found before by Des Cartes, Gassen­dus, and Sir Kenelme Digby. But let any man read Des Cartes, he shall finde, that he attri­buteth no Motion at all to the object of sense, but an inclination to action, which inclination no man can imagine what it meaneth. And for Gassendus, and S. Kenelme Digby, it is manifest by their writings, that their opinions are not different from that or Epicurus, which is very diffe­rent from mine. O [...] if these two, or any of those I conversed with at Paris, had prevented me in publi [...]ing my own D [...]trine, yet since it was there known, and declared for mine by Mer­sennus in the P [...]eface to his Ballistica (of which the three fi [...]st leaves are imployed wholly in the setting [...] of my opinion concerning sense, and the rest of the faculties of the S [...]ul) they ought not therefo [...]e to be said to have found it out before me. And consequently this answer [Page 59] which you say was given me by Vindex was nothing else but Untruth and Envy; and (because it was done by way of Visit) Incivility. But you have not alleadged, nor can alleadge any words of mine, from which can be drawn that I am so angry as you say I am with those that submit not to my Dictates. Though the discipline of the University be never so good; yet certainly this behavour of yours and his are no good Arguments to make it thought so. But you the Pro­fessor of Geometry, that out of my words spoken against Vindex in my 20 Chapter, argue my angry humour; do just as well, as when (in your Arithmetica Infinit [...]rum) from the continuall increase of the excess of the row of Squares above the third part of the aggregate of the greatest, you conclude they shall at last be equall to it. For though you knew that Vindex had given me first the wo [...]st words that possibly can be given, yet you would have that return of mine to be a Demonstration of an ang [...]y humou [...]; not then knowing what I told you even now in the be­ginning of this Lesson, of the sentence given by Vespasian. But to this Point I shail speak again hereafter.

Your third Accusation is, That I had my doctrine of Vision, which I pretend to be my own, out of papers which I had a long time in my hands of Mr. Warners. I never had sight of any of Mr. Warn. papers in all my life but that of Vision by Refraction (which by his approbation I car­ryed with me to Paris, and caused it to be printed under his own name, at the end of Mersen­nus his Cogitata Physico-Mathematica, which you may have there seen) and another Treatise of the Proportions of Alloy in Gold and Silver coine; which is nothing to the present purpose. In all my conversation wi [...]h him, I never heard him speak of any thing he had written, or was writing de Penicillo optico. And it was from me that he first heard it mentioned that Light and Colour were but Fancy. Which he imbraced presently as a truth, and told me it would re­move a rub he was then come to in the discovery of the place of the Image. If after my going hence he made any use of it (though he had it from me, and not I from him) it was well d [...]ne. But wheresoever you finde my Principles, make use of them, if you can, to demonstrate all the Symptomes of Vision; and I will do (or rather have done and mean to publish) the same; and let it be judged by that, whether those Principles be of mine, or other mens invention. I give you time enough, and this advantage besides, that much of my Optiques hath been pri­vately read by others. For I never refused to lend my papers to my friends, as knowing it to be a thing of no prejudice to the advancement of Philosophy, though it be (as I have found it since) some prejudice to the advancement of my own reputation in those Sciences; which repu­tation I have alwayes postposed to the common benefit of the studious.

You say further (you the Geometrician) that I had the Proposition of the Spirall Line equall to a Parabolicall Line from Mr. Robervall. True. And if I had remembred it, I would have ta­ken also his demonstration, though if I had publisht his, I would have suppressed mine. I was comparing in my thoughts those two Lines, Spirall and Parabolicall, by the Motions wherewith they were described; and considering those Motions as uniform, and the Lines from the Center to the Circumference, not to be little Parallelograms, but little Sectors, I saw that to compound the true Motion of that Point which described the Spirall, I must have one Line equall to half the Perimeter, the other equall to half the Diameter. But of all this I had not one word written. But being with Mersennus and Mr. Robervall in the Cloister of the Convent, I drew a Figu [...]e on the wall, and Mr. Robervall perceiving the deduction I made, told me that since the Motions which make the Parabolicall Line, are one uniform, the other accelerated, the Motions that make the Spirall must be so also; Which I presently acknowledged; and he the next day, from this very method brought to Mersennus the demonstration of their equality. And this is the [...] mentioned by Mersennus, Prop. 25. Corol. 2. of his Hydraulica; Which I know not who hath most magnanimously interpreted to you in my disgrace.

The fourth accusation is, That I have injured the Universities. Wherein? First▪ In that I [...]oul [...] have the Doctrine of my Leviathan by entire Soveraignty [...] imposed on them. You often upb [...]aid me with thinking well of my own Doctrine; and gram by consequence, that I thought this Doctrine g [...]od. I desired not therefore that any thing should b [...] imposed upon them, [Page 60] but what (at least in my opinon) was good both for the Common-wealth and them. Nay more I would have the State make use of them to uphold the civill Power, as the Pope did to uphold the Ecclesiasticall. Is it not absurdly done to call this an Injury? But to question (you will say) whether the Civill Doctrine there taught, be such as it ought to be, or not, [...]a disgrace to the Unive [...]sities. If that be certain, it is ce [...]tain also that those Se [...]mons and Books, which have been preached and published, both against the former and the present Government, direct­ly or obliquely, were not made by such Ministe [...]s and others as had their breeding in the Uni­versities; though all men know the contrary. But the Doctrine which I would have to be taught there, what is it? It is this, That all men that live in a Common-wealth, and receive protection of their lives and fortunes from the Supreme Governour thereof, are reciprocally [...]ound as far as they are able, and shall be required, to protect that Governour. Is it, think you, an un [...]e [...]sonable thing to impose the teaching of such Doctrine upon the Universities? O [...] will you say they taught it before, when you know that so many men which came from the Universities to preach to the People, and so many others that were not Ministers did stir the People up to resist the then Supreme Civill Power? And was it not truly therefore said, that the Universities receiving their Discipline from the Authority of the Pope, were the Shops, and Operatories of the Clergy? Though the Competition of the P [...]pall and Civill power be ta­taken now away, yet the Competition between the Ecclesiasticall and the Civill power hath manifestly enough appeared very lately. But neither is this an upbraiding of an University (which is a Corporation or Body Artificial) but of particular men that desire to uphold the Au­thority of a Church, as of a distinct thing from the Common-wealth. How would you have exclaimed, it instead of [...]ecommending my Lev [...]athan to be taught in t [...]e Universities, I had recommen [...] the errecting of a New and Lay-University, whe [...]ein Lay-men should have the reading of Physiqu [...]s, Mathematicks, Morall Philosophy, and Politicks, as the Clegy have now the sole teaching or Divinity? Yet the thing would be profitable, and tend much to the Polishing of ma [...]s [...]ature, without much publique charge. There will need but one House, and the endowment of a few Professions. And to make some learn the better, it would do very well that none should come thither sent by their Parents, as to a Trade to get their living by, but that it should be a place for such ingenuous men, as being free to dispose of their own time, love Truth for it self. In the mean time Divinity may go on in Oxford and Cambridge to furnish the Pulpit with men to cry down the Civil Power, if they continue to do as they did. If I had (I say) made such a Motion in my Leviathan, though it would have offended the Divines, yet it had been no injury. But 'tis an injury (you will say) to deny in generall the utility of the Antient Schooles, and to deny that we have received from them our Geometry. True, if I had not spoken distinctly of the Schools of Philosophy, and said expresly, that the Geometricians passed n [...]t then under the name of Philosophers; and that in the School of Plato (the best of the Antient Philosophers) none were received that were not alrea [...]y in some measure Geome­tricians. Euclide taught Geometry; but I never heard of a Sect of Philosophers, called Euclidians, or Alexandrians, or ranged with any of the other Sects, as Peripat [...]tiques, Stoiques, Academiques, Epicureans, Pyrrhonians, &c. But what is this to the Universities of Christendome? Or why are we beholding for Geometry to our Universities, more then to Gresham Colledge, or to private men in London, Paris, and other places, which never taught or learned it in a publique School? For even those men that living in our Universities have most advanced the Mathematiques, attained their knowledge by other means then that of publique Lectures, where few Auditors, and those of unequall proficiency cannot make benefit by one and the same Lesson. And the true use or publique Professors, especially in the Mathematiques, being to resolve the Doubts, and Problems (as far as they can) of such as come unto them with desire to be informed.

That the Universities now are not regulated by the Pope, but by the Civill Power, is true, and well. But where say I the contrary? And thus much for the first in­jurie.

[Page 61] Another (you say) is this, that in my Leviathan Pag. 380. I say, The principall Schooles were or [...]a [...]n [...]d for the three Professions of Roman Religion, Roman Law, and the Art of Me­dicine. Thirdly, that I say. Philosophy had no otherwise place there then as a hand-maid to Roman Religion, Fourthly, since the authority of Aristotle was received there, that Stu­dy is not pr [...]perly Philosophy but A [...]ist [...]telity. Fifthly, That for Geometry, till of late times it had no place there at all. As for the second, it is too evident to be denyed; the Fellowships having been all ordained for those Professions; and (saving the Change of Religion) being so yet. Nor hath this any Reflection upon the Universities, either as they now are, or as they then we [...]e, seeing it was not in their own power to endow themselves, or to receive other Laws and Discipline then their Founder, and the State were pleased to ordain. For the third, it is also evident. For all men know that none but of the Roman Religion had any Stipend or pre [...]e [...]ment in any University, where that Religion was established; No, nor for a great w [...]ile, in their Common-wea [...]s; but were every where persecuted as H [...]r [...]tiques. But you will say the words in my L [...]viat [...]an are not, Philosophy ha [...] no place, but hath no place. Are you not ashamed [...] to [...]y charge a mistake or the word hath [...]o [...] had? Which was either a mi­stake of the Printer, or i [...] it [...] so in the Copy, it could be no other then the mistake of a let­ter in the writing, unless you think you ca [...] [...]ke m [...]n believe that after fifty years being ac­quainted with what was publiquely p [...]of [...]st an [...] practised in Ox [...]rd and Cambridge, I knew not what Religion they were of. [...] taking [...] advantage from the mistake [...] a word, or of a l [...]t [...]er, I finde also in the Elenchus, whe [...] for praetendit s [...]s [...]ire, there is praetendit s [...]ire, wh [...] you the Geome [...]ician [...] [...]mble, mistaking it I think for an Anglicisme, not for a fault in the impression.

To the [...]o [...]th, you p [...]ete [...]d, that men are not now so tied to Arist [...]tle as not to enjoy a liberty of Philosop [...]i [...]g, [...] it were otherwise when I was conversant in Magdalen Hall. Was it so then? [...]hen am I absolved, unless you can shew some publique A [...] of the University made since that time to alte [...] it. For it is not enough to name some few particular ingenuous men that usu [...]p [...] [...] Liberty in their private discourses, or (with connivence) in their p [...]b­like disputatio [...]s. And your Doctrine, that even here you avow, of Abstracted Essences, Immatertall Substances, and of Nunc-stans; and your improper language in using the word (not as mine, for I have it no where) Successive Eternity; as also your Doctrine of Conden­sation, and your a [...]guing from naturall reason the incomp [...]eh [...]nsible Mysteries of Religion, and your Malicious W [...]iting, are very [...]wd [...]gnes, that you your selves are none of those which you say do freely Philosophise, but that both your Philosophy and your Language are under the S [...]rvitude, not of the Roman Religion, but of the Ambition of some other Docto [...]s, that seek, as the Romish Clergy did, to draw all humane learning [...] the upholding of their Power Ecclesiasticall. Hitherto therefore there is no injury done to t [...]e Universiti [...] [...]or [...] fifth, you grant it, namely, That till of late there was no allowance for the [...]ing o [...] G [...]ry. But least you [...]houl [...] be thought to grant me any thing, y [...]u say, you th [...] [...], G [...]ome [...]ry hath now s [...] much plac [...] in the Universities, that when Mr. [...]bs shall [...]au [...] published his Phi [...] ­sophicall an [...] Geometricall p [...]s, you assur [...] your [...] you shall be [...] [...] [...] g [...]eat [...]umb [...]er in the University, who will un [...]rst [...]n [...] as much or more of them, [...] [...]esireth [...] [...]ul [...], &c. But though this be [...] o [...] t [...] now, yet it maketh nothing against my [...]. I k [...]w [...] enough that Sir He [...]ry S [...]iles [...]ectu [...]s w [...]e rounde [...] an [...] e [...]owed [...] Di [...] I [...]ny th [...]n, th [...]t there were in Oxford [...]any [...] Geometricians? But I [...]ny now, [...]at [...] or you is or the Number. For my Philosophicall and G [...] pi [...]es, [...] [...], and you have understood onely so muc [...] in them, as all [...] will easily [...] by your o [...]jectio [...]s to them, and by your own pub [...] Geometry, that neither or you understand any [...] [...] in P [...]ilos [...] ­phy or in [...]. And yet you woul [...] have [...] Books o [...] yours [...]o [...] an A [...]g [...]en [...], and to be an Ind [...]x o [...] the Philosophy and Geomet [...]y [...] [...]e [...]ound in the [...]. W [...] is a greater injury and di [...]g [...]ace to them, th [...] any wo [...]s o [...] mine though [...] by [...]o [...]r selves.

[Page 62] Your last and greatest accusation, or rather railing (for an Accusation should contain, whe­ther true or false, some particular fact, or certain words, out of which it might seem at least to be inferred) is, that I am an enemy to Religion. Your words are, It is said that Mr Hobs is no otherwise an Enemy to the Roman Religion, saving onely as it hath the name of Religion. This is said by Vindex. You the Geometrician in your Epistle Dedicatory say thus, With what pride and imperiousness be tramples on all things both Humane and Divine, uttering fearfull and horrible words of God, (peace) of Sin, of the Holy Scripture, of all Incorporeal substances in generall, of the Immortall soul of man, and of the rest of the weighty points of Religion (down) it is not so much to be doubted as lamented. And at the end of your objecti­ons to the 18 Chapter, Perhaps you take the whole History of the Full of Adam for a Fable, which is no wonder, wh [...] you say the Rules of honouring and worshipping of God are to be taken from the Laws. Down I say; you ba [...]ke now at the Supreme Legislative Power. Therefore it is not I, but the Laws which must rate you off. But do not many other men as well as you read my Leviathan, and my other Books? And yet they all finde not such enmity in them against Religion. Take heed of calling them all Atheists that have read and approved my Levi­athan. Do you think I can be an Atheist and not know it? Or knowing it durst have offer­ed my Atheism to the Press? Or do you think him an Atheist, or a co [...] of the Holy Scripture, that sayeth nothing of the Deity, but what he prov [...] by the Scripture? You that take so hainously that I would have the Rules of Gods worship [...] a Christian Common-wealth taken from the Laws, tell me, from whom you would have them taken? From yourselves? Why so, more then from me? From the Bishops? Right, if the Supreme Power of the Common-wealth will have it so; If not, why from them rather then from me? From a Consistory of Presbyters by themselves, or joyned with Lay-Elders, whom they may sway as they please? Good, If the Supreme Governour of the Common-wealth will have it so. If not, why from them, rather then from me, or from any man else? They are wiser and learn [...]der then I. It may be so; but it has not yet appeared. Howsoever, let that be granted. Is there any man so very a fool as to subject himself to the Rules of other men in those [...]ings which so neerly con­cern himself, for the Title they assume of being wise and learned, unless they also have the sword which must protect them. But it seems you understand the sword as comprehended. If so, do you not then receive the Rules of Gods worship from the Civill Power? Yes doubtless; and you would expect, if your Consistory had that sword, that no man should dare to exercise o [...] teach any Rules concerning Gods worship which were not by you allowed. See therefore how much you have been transported by your malice towards me, to injure the Civill Power by which you live. If you were not despised, you would in some places and times, where and when the Laws are more severely executed, be shipt away for this your madness to America, I would say, to Anticyra. What luck have I, when this, of the Laws being the Rules of Gods publique worship, was by me said and applyed to the Vindication of the Church of England [...] the Power of the Roman Clergy, it should be followed with such a storm from the Ministers Presby­terian and Episcopall of the Church of England? Again, for those other Points, namely, that I approve not of Incoporeall Bodies, not of other Immortality of the soul, then that which the Scripture calleth Eternall Life, I do but as the Scripture leads me. To the Texts whereof by me alleadged, you should either have answered, or [...] forborne to revile me for the conclusions I derived from them. Lastly, what an absurd question is it to ask me whether it he in the Power of the Magistrate, whether the world be eternall or not? It were fit you knew [...] in the Power of the Supreme Magistrate to make a Law for the punishment of them that shall pronounce publiquely of that question any thing contrary to that which the Law hath once pronounced. The truth is, you are content that the Papall power be cut off, and declaimed against as much as any man will; but the Ecclesiasticall Power which of la [...]e was ai [...]ed at by the Clergy here, be­ing a part thereof, every violence done to the Papall Power is sensible to them yet; like that which I have heard say of a man, whose leg being cut off for prevention of a Gangrene that began in his Toe, would nevertheless complain of a pain in his Toe, when his leg was cut off.

[Page 63] Thus much in my defence; which I believe if you had foreseen, this Accusation of yours had been left out. I come now to examine (though it be done in part already) what Manners those are which I finde every where in your Writings.

And first, How came it into your minds that a man can be an Atheist, I mean an Atheist in his Conscience? I know that David confesseth of himself, upon sight of the Prosperity of the wicked, that his feet had almost slipt, that is, that he had slipt into a short doubt­fulness of the Divine Providence. And if any thing else can cause a man to slip in the same kinde, it is the seeing such as you (who though you write nothing, but what is dictated to each of you by a Doctor of Divinity) to break the greatest of Gods Commandement (which is Cha­rity) in every Line before his face. And though such forgettings of God be somewhat more then shor [...] doubtings, and sudden transportations incident to humane Passion, yet I do not for that cause think you Atheists, and enemies of Religion, but onely ignorant and imprudent Chri­stians. But how, I say, could you think me an Atheist, unless it were because finding your doubts of the Deity more frequent then other men do, you are thereby the apter to fall upon that kinde of reproach? Wherein you are like women of poor & evil Education when they scold; amongst whom the readiest disgracefull word is Whore. Why not Thief, or any other ill name, b [...]t because when they remember themselves, they think that reproach the likeliest to be true?

Secondly, tell me what crime it was which the Latines called by the name of Scelus? You think not (unless you be Stoiques) that all Crimes are equall. Scelus was never used but for a Crime of greatest mischief, as the taking away of Life and Honour; and besides, basely acted, as by some clandestine way, or by such a way as might be covered with a Lve. But when you insi­nuate in a writing publisht that I am an Atheist, you make your selves Authors to the multitude, and do all you can to stir them up to attempt upon my life; and if it succeed, then to sneak out of it by leaving the fault on them that are but actors. This is to indeavour great mischief basely; and therefore Scelus. Again, to deprive a man of the honour he hath merited, is no little wickedness; and this you endeavour to do by publishing falsely that I challenge as my own the in­ventions of other men. This is therefore Scelus Publiquely to tel all the world that I will be angry with all men that do not presently submit to my Dictates; to deprive me of the friendship of [...] the world. Great dammage, and a lie, and yours. For to publish any untruth of another man to his disgrace, on hear-say from his enemy, is the same fault as if he publisht it on his own credit. I [...] I should say I have heard that Dr. Wallis was esteemed at Oxford for a simple fellow, and much interiour to his fellow-professor Dr. War [...] (as indeed I have heard, but do not believe it) though this be no great disgrace to Dr. Wallis, yet he would think I did him injury. There­fore publique Accusation upon hear-say is Scelus. And whosoever does any of these things does Seel rate. But you the Professors of the Mathematiques at Oxford, by the advice of two Doctors of Divinity have dealt thus with me. Therefore you have done (I say not foolishly, though no wickedness he without folly, but Scelera [...], [...].

Thirdly, it is ill Manners, in reprehending trut [...], to send a man in a boasting way to your own errors; as you the Professor of Geometry have often sent me to your two T [...]actates of the Angle of Contact, and Arithmetica Infinitorum.

Fourthly, it is ill Manners, to diminish the just reputation of worthy men after they be dead, as you the Professor of Geometry have done in the case of Joseph Scaliger.

Fifthly, when I had in my Leviathan suffered the Clergy of the Church of England to [...]pe, you did imprudently in bringing any of them in again. An Ulysses upon so light an occasion would not have ventured to [...] again into the Cave of Polyphemus.

Lastly, how ill does such Levi [...]y and Scurrility, which both of you have shewn so often in your writings, become the gravity and sanctity requisite to the calling o [...] the Ministery? They are too many to be repeated. Do but consider you the Geometrician how unhandsome it is to play upon my name, when both yours and mine are pl [...]bcian names; though from Willis by Wallis, you go from yours in Wallisius. The [...]est of using at every word M [...] Hobb [...], is lest to them beyond Sea. But this is not to ill as some of the rest. I will write out one of them as it is in [Page 64] the fourth Page of your Elenchus; Whence it appears that your Empusa was of the number of those Fairies which you call in English Hob-goblins. The word is male of [...] and [...]; and [...]hence comes the childrens play called the play of Empusa, Anglicé (hitherto in Latine all but Hob-goblins, then follows) Fox, Fox, come out of your hole (in English, then in Latine again,) in which the boy that is called the Fox, holds up one foot, and jumps with the other, which in English is to hop. When a stranger shall read this, and hoping to finde therein some witty conceit, shall with much adoe have gotten it interpreted and explained to him, what will he think of our Doctors of Divinity at Oxford, that will take so much pains as to go out of the language they set forth in, for so ridiculous a purpose? You will say it is a pretty Paranomafia. How you call it there I know not, but it is common­ly called here a Clinch; and such a one as is too insipide for a boy of twelve years old, and very unfit for the sanctity of a Minister, and gravity of a Doctor of Divinity. But I pray you tell me where it was you read the word Empusa for the boys play you speak of, or for any other play amongst the Greeks. In this (as you have done throughout all your other writings) you presume too much upon your first cogitations. There be a hundred other scoffing passages, and ill-favou­red attributes given me in both your writings, which the Reader will observe without my poin­ting to them, as easily as you would have him; and which perhaps some young Students, finding them full of Gall, will mistake for Salt. Therefore to disabuse those youngmen, and to the end they may not admire such kind of wit, I have here and there been a little [...] with you then else I would have been. If you think I did not spare you, but that I had not wit enough to give you as scornfull names as you give me, are you content I should try? Yes (you the Geo­metrician will say) give me what names you please, so you call me not Arithmetica Infinito­rum (I will not.) Nor Angle of Contact, nor Arch-Spirall, Nor Quotient (I will not.) But I here dismiss you both together. So go your wayes, you uncivill Ecclesia, [...]ques, Inhumane Divines, Dedoctors of Morality, Unafinous Colleagues, Egregious pa [...]r of Iss [...], most wretched Vindices and Indices Academiarum; and remember Vespasians Law, that it is uncivill to give ill language first, but civill and lawfull to return it. But much more remember the Law of God, to obey your Soveraigns in all things; and not only not to dero [...] from them, but also to pray for them, and as far as you can to maintain their Authority, and therein your own [...]. [...] (do you hear?) take heed of speaking your minde so cleerly in answering m [...] Leviathan, as I have done in writing it. You should do best not to meddle with it at all, because it is underta­ken, and in part published already, and will be better performed, from Term to Term, by one Christopher Pike.

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